Compressible Fluid Equations Research Articles

Dissipation Limit for the Compressible Navier-Stokes to Euler Equations in One-Dimensional Domains

We prove that as the viscosity and heat-conductivity coefficients tend to zero, respectively, the global solution of the Navier-Stokes equations for one-dimensional compressible heat-conducting fluids with centered rarefaction data of small strength converges to the centered rarefaction wave solution of the corresponding Euler equations uniformly away from the initial discontinuity.

Consistent equations for open-channel flows in the smooth turbulent regime with shearing effects

Consistent equations for turbulent open-channel flows on a smooth bottom are derived using a turbulence model of mixing length and an asymptotic expansion in two layers. A shallow-water scaling is used in an upper – or external – layer and a viscous scaling is used in a thin viscous – or internal – layer close to the bottom wall. A matching procedure is used to connect both expansions in an overlap domain. Depth-averaged equations are then obtained in the approximation of weakly sheared flows which is rigorously justified. We show that the Saint-Venant equations with a negligible deviation from a flat velocity profile and with a friction law are a consistent set of equations at a certain level of approximation. The obtained friction law is of the Kármán–Prandtl type and successfully compared to relevant experiments of the literature. At a higher precision level, a consistent three-equation model is obtained with the mathematical structure of the Euler equations of compressible fluids with relaxation source terms. This new set of equations includes shearing effects and adds corrective terms to the Saint-Venant model. At this level of approximation, energy and momentum resistances are clearly distinguished. Several applications of this new model that pertains to the hydraulics of open-channel flows are presented including the computation of backwater curves and the numerical resolution of the growing and breaking of roll waves.

Biquaternionic reformulation of multifluid plasma equations

Extending the biquaternionic generalization of the Maxwell type equations of compressible fluids, an effort has been introduced for the reformulation of analogous multifluid plasma equations. With the help of an exact correlation between the equations of classical electrodynamics and multifluid plasma model, suitable and elegant biquaternionic expressions including corresponding matrix representations have been presented. The proposed model in this paper generalizes Maxwell type equations of multifluid plasma in a simple, compact and consistent manner. Finally, using the advantages of biquaternions, the wave equation for plasma is reformulated similarly to the compressible fluids.

Arbitrary-Lagrangian–Eulerian Discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes

We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes, like molecular viscosity or heat conduction. High order piecewise polynomials of degree N are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach, making use of an element-local space–time Galerkin finite element predictor. A novel nodal solver algorithm based on the HLL flux is derived to compute the velocity for each nodal degree of freedom that describes the current mesh geometry. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Each technique generates a corresponding number of geometrical degrees of freedom needed to describe the current mesh configuration and which must be considered by the nodal solver for determining the grid velocity.The connection of the old mesh configuration at time tn with the new one at time tn+1 provides the space–time control volumes on which the governing equations have to be integrated in order to obtain the time evolution of the discrete solution. Our numerical method belongs to the category of so-called direct Arbitrary-Lagrangian–Eulerian (ALE) schemes, where a space–time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry (including a possible rezoning step) directly during the computation of the numerical fluxes. We emphasize that our method is a moving mesh method, as opposed to total Lagrangian formulations that are based on a fixed computational grid and which instead evolve the mapping of the reference configuration to the current one.Our new Lagrangian-type DG scheme adopts the novel a posteriori sub-cell finite volume limiter method recently developed in [62] for fixed unstructured grids. In this approach, the validity of the candidate solution produced in each cell by an unlimited ADER-DG scheme is verified against a set of physical and numerical detection criteria, such as the positivity of pressure and density, the absence of floating point errors (NaN) and the satisfaction of a relaxed discrete maximum principle (DMP) in the sense of polynomials. Those cells which do not satisfy all of the above criteria are flagged as troubled cells and are recomputed at the aid of a more robust second order TVD finite volume scheme. To preserve the subcell resolution capability of the original DG scheme, the FV limiter is run on a sub-grid that is 2N+1 times finer compared to the mesh of the original unlimited DG scheme. The new subcell averages are then gathered back into a high order DG polynomial by a usual conservative finite volume reconstruction operator.The numerical convergence rates of the new ALE ADER-DG schemes are studied up to fourth order in space and time and several test problems are simulated in order to check the accuracy and the robustness of the proposed numerical method in the context of the Euler and Navier–Stokes equations for compressible gas dynamics, considering both inviscid and viscous fluids. Finally, an application inspired by Inertial Confinement Fusion (ICF) type flows is considered by solving the Euler equations and the PDE of viscous and resistive magnetohydrodynamics (VRMHD).

Global existence and optimal convergence rates of solutions for 3D compressible magneto-micropolar fluid equations

The Cauchy problem for the three-dimensional compressible magneto-micropolar fluid equations is considered. Existence of global-in-time smooth solutions is established under the condition that the initial data are small perturbations of some given constant state. Moreover, we obtain the time decay rates of the higher-order spatial derivatives of the solution by combining the Lp-Lq estimates for the linearized equations and the Fourier splitting method, if the initial perturbation is small in H3-norm and bounded in L1-norm.

Spacetime algebra for the reformulation of fluid field equations

In the light of the analogy between electromagnetism and fluid dynamics, the Maxwell-type equations of compressible fluids are reformulated on the basis of spacetime algebra. In this paper, it is proved that this algebra provides an efficient mathematical tool for describing fluid fields in a compact and elegant way. Moreover, the fluid wave equation in terms of potentials are derived in a form similar to electromagnetic and gravitational counterparts previously derived using spacetime algebra.

Cell viability and shock wave amplitudes in the endothelium of porcine cornea exposed to ultrashort laser pulses

Some forms of keratoplasty assisted by ultrashort-pulse lasers require performing laser cuts close to the endothelium, which requires the knowledge of "safe" values concerning incision depth and pulse energy preserving endothelial cell viability. Our study aims to determine the thresholds for cell death in porcine corneas exposed to ultrashort laser pulses, in terms of laser pulse energy and nearness of the impacts to the endothelium. Using a laboratory laser set-up, lamellar cuts were induced while varying pulse energies and distances from the endothelium. A fluorescent staining protocol was used to determine the percentage of surviving endothelial cells. Numerical simulations of the Euler equations for compressible fluids provided pressure level and axial and radial pressure gradient estimates at the endothelium. Ninety percent of the endothelial cells survived when using 16.5μJ pulses no closer than 200μm to the endothelium, or pulses not exceeding 2μJ at a distance of 50μm. The comparison of the observed percentage of surviving cells with the estimates of the shock wave amplitudes and gradients generated by the laser pulses yielded cell death thresholds at amplitudes in the megapascal range, or gradients of the order of 108Pa/m. Our results provide limits in terms of pulse energy and distance of the incision from the endothelium within which endothelial cell viability is preserved. Current forms of corneal laser surgery are compatible with these limits. However, these limits will need to be considered for the development of future laser routines working in close proximity to the endothelium.

Application of the Hierarchical Functions Expansion Method for the Solution of the Two Dimensional Navier-Stokes Equations for Compressible Fluids in High Velocity

This work presents a new application for the Hierarchical Function Expansion Method for the solution of the Navier-Stokes equations for compressible fluids in two dimensions and in high velocity. This method is based on the finite elements method using the Petrov-Galerkin formulation, know as SUPG (Streamline Upwind Petrov-Galerkin), applied with the expansion of the variables into hierarchical functions. To test and validate the numerical method proposed as well as the computational program developed simulations are performed for some cases whose theoretical solutions are known. These cases are the following: continuity test, stability and convergence test, temperature step problem, and several oblique shocks. The objective of the last cases is basically to verify the capture of the shock wave by the method developed. The results obtained in the simulations with the proposed method were good both qualitatively and quantitatively when compared with the theoretical solutions. This allows concluding that the objectives of this work are reached.

Spectral analysis for travelling waves in compressible two-phase fluids of Navier–Stokes–Allen–Cahn type

This is the first part of two papers whose purpose is to investigate stability of travelling wave solutions to the so-called Navier–Stokes–Allen–Cahn system. This set of equations is a combination of the Navier–Stokes equations for compressible fluids supplemented with a phase field description of Allen–Cahn type. The main part of this work deals with studying the problem obtained by linearizing the NSAC system around so-called standing waves. The main results are (1) local well-posedness of the linearized equations and (2) a detailed description of the point and essential spectrum. As a by-product, we obtain analyticity of the associated semigroup.

СЕКЦИОНИРОВАНИЕ НАБЕГАЮЩЕГО ПОТОКА ПРИ СВЕРХЗВУКОВОМ ПОПЕРЕЧНОМ ОБТЕКАНИИ КРУГОВОГО ЦИЛИНДРА

The numerical simulation results of supersonic incident flow along a cross circular cylinder of different length in Mach number range from 2.5 up to 3.5 with Reynolds number Re= 2∙10 6 are presented in the article. Numerical simulation was carried out using unsteady three-dimensional Reynolds-averaged Navier–Stokes equations for compressible fluid. For the equation system closure four turbulence models such as SST (Shear Stress Transport), SAS SST (Scale-Adaptive Simulation Shear Stress Transport), DES (Detached Eddy Simulation) and LES (Large Eddy Simulation) were used. The numerical solution of the reference equations was obtained due to a control volume approach. In the calculations numerical methods that have the second order of approximation by space variables were used. The solution is presented to the task in three dimensional statement of flow around the infinite cylinder. Due to the results of the numerical modeling the availability of large-scale vortexes that bring to a sectioning of the incident flow was discovered. In process of streamlining the flow is divided into equal segments (sections) along the cylinder. Formed sections of the flow are structured along the cylinder and have the equal scale. The influence of the cylinder length on a sectioning scale and structure is considered. The numerical simulation results of the supersonic incident flow sectioning along a cross cylinder with the length 6d, 4d and 2d (where d is a diameter of current cylinder) are presented in the article. In this case the size of control volumes along the cylinder is constant. In addition Mach number influence on the sectioning scale and structure is considered. Mach number effect on origin and attenuation is done for the cylinder length 4d in Mach number range from 2.5 up to 3.5 with the constant Reynolds number. Also the causes of sectioning generating of the supersonic incident flow along the cross cylinder are discussed. The possible causes are space discontinuity of incident flow, flow instability in an area of the front critical point and interaction between the shock wave and vortex flow in a shock layer.

A new model for the reformulation of compressible fluid equations

Stimulating the biquaternionic generalization of electric and magnetic fields in electromagnetism, a new model has been proposed to present the Maxwell type equations of compressible fluids. The fluid wave equation has been expressed in a compact and elegant manner. Similarly, the generalized Poynting theorem for fluids has been derived analogous to electromagnetism and linear gravity. Moreover, a brief survey on biquaternionic matrices and the corresponding matrix representations of fluid equations have been given.

A three-equation model for thin films down an inclined plane

We derive a new model for thin viscous liquid films down an inclined plane. With an asymptotic expansion in the long-wave limit, the Navier–Stokes equations and the work–energy theorem are averaged over the fluid depth. This gives three equations for the mass, momentum and energy balance which have the mathematical structure of the Euler equations of compressible fluids with relaxation source terms, diffusive and capillary terms. The three variables of the model are the fluid depth, the average velocity and a third variable called enstrophy, related to the variance of the velocity. The equations are numerically solved by classical schemes which are known to be reliable and robust. The model gives satisfactory results both for the neutral stability curves and for the depth profiles of wavy films produced by a periodical forcing or by a random noise perturbation. The numerical calculations agree fairly well with experimental measurements of Liu & Gollub (Phys. Fluids, vol. 6, 1994, pp. 1702–1712). The calculation of the wall shear stress below the waves indicates a flow reversal at the first depth minimum downstream of the main hump, in agreement with experiments of Tihon et al. (Exp. Fluids, vol. 41, 2006, pp. 79–89).

Direct simulation of the turbulent boundary layer on a plate

A numerical method for the integration of three-dimensional Navier–Stokes equations for compressible fluid as applied to direct numerical simulation is proposed. By way of example, the boundary layer on a plate is simulated. The computations were carried out for Reθ = 1500. The computational grid consisted of a half billion nodes. The flow region includes the laminar, transitional, and turbulent zones. The numerically obtained distributions of average velocity, friction, and pulsations are compared with experimental data and available numerical solutions.

On the large time behavior of the electromagnetic fluid system in [formula omitted

This paper is concerned with the global existence and time-asymptotic behavior of solutions to the three dimensional complete electromagnetic fluid system (or Maxwell–Navier–Stokes equations) for viscous compressible fluids. The global classical solution is established when the initial data are small perturbations of some given constant state. Moreover, the optimal decay estimates of the solution in Lq with 2≤q≤6 and its higher-order spatial derivatives in L2 are also obtained.

Boundary-layer phenomena for the cylindrically symmetric Navier–Stokes equations of compressible heat-conducting fluids with large data at vanishing shear viscosity

This paper concerns the asymptotic behavior of the solution to an initial-boundary value problem of the cylindrically symmetric Navier–Stokes equations with large data for compressible heat-conducting ideal fluids, as the shear viscosity μ goes to zero. A suitable corrector function (the so-called boundary-layer type function) is constructed to eliminate the disparity of boundary values. As by-products, the convergence rates of the derivatives in L2 are obtained and the boundary-layer thickness (BL-thickness) of the value with is shown by an alternative method, compared with the results proved in Jiang and Zhang (2009 SIAM J. Math. Anal. 41 237–68) and Qin et al (2015 Arch. Ration. Mech. Anal. 216 1049–86).

Mixing rules and the Navier-Stokes-Cahn-Hilliard equations for compressible heat-conductive fluids

The framework of this article is the compressible Navier-Stokes-Cahn- Hilliard system for the dynamics of a fluid whose two phases are macroscopically immiscible; partial mixing is permitted leading to narrow transition layers. This so-called NSCH model was originally derived by Lowengrub and Truskinowsky [7], but only for the isothermal case. The purpose of this work is to present the non-isothermal version as well as a well-posedness result. The PDEs constitute a strongly coupled hyperbolicparabolic system.

Computational study of shock waves propagating through air-plastic-water interfaces

The following study ismotivated by experimental studies in traumatic brain injury (TBI). Recent research has demonstrated that low intensity non-impact blast wave exposure frequently leads to mild traumatic brain injury (mTBI); however, the mechanisms connecting the blast waves and the mTBI remain unclear. Collaborators at the Seattle VA Hospital are doing experiments to understand how blast waves can produce mTBI. In order to gain insight that is hard to obtain by experimental means, we have developed conservative finite volume methods for interface-shock wave interaction to simulate these experiments. A 1D model of their experimental setup has been implemented using Euler equations for compressible fluids. These equations are coupled with a Tammann equation of state (EOS) that allows us to model compressible gas along with almost incompressible fluids or elastic solids. A hybrid HLLC-exact Eulerian-Lagrangian Riemann solver for Tammann EOS with a jump in the parameters has been developed. The model has shown that if the plastic interface is very thin, it can be neglected. This result might be very helpful to model more complicated setups in higher dimensions.

Incompressible limit of solutions of multidimensional steady compressible Euler equations

A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompresibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.

Finite Volume Method for the Relativistic Burgers Model on a (1+1)-Dimensional de Sitter Spacetime

Several generalizations of the relativistic models of Burgers equations have recently been established and developed on different spacetime geometries. In this work, we take into account the de Sitter spacetime geometry, introduce our relativistic model by a technique based on the vanishing pressure Euler equations of relativistic compressible fluids on a (1+1)-dimensional background and construct a second order Godunov type finite volume scheme to examine numerical experiments within an analysis of the cosmological constant. Numerical results demonstrate the efficiency of the method for solutions containing shock and rarefaction waves.

Numerical Methods for Interface Coupling of Compressible and Almost Incompressible Media

Many experiments in biomedical applications and other disciplines use a shock tube. These experiments often involve placing an experimental sample within a fluid-filled container, which is then placed inside the shock tube. The shock tube produces an initial shock that propagates through gas before hitting the container with the sample. In order to gain insight into the shock dynamics that is hard to obtain by experimental means, computational simulations of the shock wave passing from gas into a thin elastic solid and into a nearly incompressible fluid are developed. It is shown that if the solid interface is very thin, it can be neglected, simplifying the model. The model uses Euler equations for compressible fluids coupled with a Tammann equation of state (EOS) to model both compressible gas and almost incompressible materials. A three-dimensional (2D axisymmetric) model of these equations is solved using high-resolution shock-capturing methods, with newly developed Riemann solvers and limiters. The methods are extended to work on a mapped grid to allow more complicated interface geometry, and they are adapted to work with adaptive mesh refinement (AMR) for higher resolution and faster computations. The Clawpack software is used to implement the method. These methods were initially inspired by shock tube experiments to study the injury mechanisms of traumatic brain injury (TBI).