Euler Flow Research Articles

Stability of stationary subsonic compressible Euler flows with mass-additions in two-dimensional straight ducts

We show existence, uniqueness and stability for a family of stationary subsonic compressible Euler flows with mass-additions in two-dimensional rectilinear ducts, subjected to suitable time-independent multi-dimensional boundary conditions at the entrances and exits. The stationary subsonic Euler equations consist a quasi-linear system of elliptic-hyperbolic composite-mixed type, while addition-of-mass destructs the usual methods based upon conservation of mass and Lagrangian coordinates to separate the elliptical and hyperbolic modes of the system. We establish a new decomposition and nonlinear iteration scheme to overcome this major difficulty. It reveals that mass-additions introduce very strong interactions in the elliptic and hyperbolic modes, and lead to a class of second-order elliptic equations with multiple integral nonlocal terms. The linearized problem is solved by studying algebraic and analytical properties of infinite weakly coupled boundary-value problems of ordinary differential equations, each with multiple nonlocal terms, after applications of Fourier analysis methods.

Efficient hyperreduction of high-order discontinuous Galerkin methods: Element-wise and point-wise reduced quadrature formulations

We develop and assess projection-based model reduction methods for high-order discontinuous Galerkin (DG) discretizations of parametrized nonlinear partial differential equations with applications in aerodynamics. Our emphasis is on the choice of hyperreduction methods. We analyze computational complexity and use numerical examples to show that typical hyperreduction methods based on element-wise or index-wise sampling of the residual vector cannot effectively reduce high-order DG discretizations, with a large number of degrees of freedom and quadrature points per element and conversely a small number of elements. To overcome this limitation, we devise a hyperreduction method for high-order DG methods that samples individual quadrature points, instead of elements, to provide a finer decomposition of the residual. We compare the formulation, implementation, and computational complexity of the element-wise and point-wise formulations. We then numerically assess the two formulations using Euler flow over a shape-parametrized airfoil and laminar and turbulent Navier-Stokes flow over a three-dimensional aerodynamic body. In each case, we observe that the point-wise hyperreduced model reduces the computational time by several orders of magnitude relative to the high-order DG method and a few orders of magnitude relative to the element-wise hyperreduced model. In addition, the computational performance of the point-wise formulation does not deteriorate with the DG polynomial degree, unlike for the element-wise formulation.

Characterization of Three-Dimensional Euler Flows Supported on Finitely Many Fourier Modes

Recently, the Nash-style convex integration has been becoming the main scheme for the mathematical study of turbulence, and the main building block of it has been either Beltrami flow (finite mode) or Mikado flow (compactly supported in the physical side). On the other hand, in physics, it is observed that turbulence is composed of a hierarchy of scale-by-scale vortex stretching. Thus our mathematical motivation in this study is to find another type of building blocks accompanied by vortex stretching and scale locality (possibly finitely many Fourier modes). In this paper, we give a complete list of solutions to the 3D Euler equations with finitely many Fourier modes, which is an extension of the corresponding 2D result by Elgindi-Hu-\v{S}ver\'ak (2017). In particular, we show that there is no 3D Euler flows with finitely many Fourier modes, except for stationary 2D-like flows and Beltrami flows. We also discuss the case when viscosity and Coriolis effect are present.

Structural stability of transonic shock flows with external force in a divergent 3-D axisymmetric perturbed nozzle

This paper concerns with the structural stability of the transonic shock for steady Euler flows with external force in a divergent 3-D axisymmetric perturbed nozzle. We first establish the existence and uniqueness of spherical symmetric transonic shock solution to steady isentropic Euler system with external force by prescribing suitable exit pressure at the exit of a straight divergent nozzle when the incoming flow at the entrance is spherical symmetric supersonic flow. Then we investigate the stability of this spherical symmetric transonic shock solution to the 3-D axisymmetric steady isentropic Euler system with external force when the incoming supersonic flow and the nozzle boundary and the exit pressure and the external force are suitable perturbed. One of new ideas in the analysis is to use the deformation-curl decomposition provided in [21] to deal with the transonic shock problem.

University of Warsaw Lagrangian Cloud Model (UWLCM) 2.0: adaptation of a mixed Eulerian–Lagrangian numerical model for heterogeneous computing clusters

Abstract. A numerical cloud model with Lagrangian particles coupled to an Eulerian flow is adapted for distributed memory systems. Eulerian and Lagrangian calculations can be done in parallel on CPUs and GPUs, respectively. The fraction of time when CPUs and GPUs work simultaneously is maximized at around 80 % for an optimal ratio of CPU and GPU workloads. The optimal ratio of workloads is different for different systems because it depends on the relation between computing performance of CPUs and GPUs. GPU workload can be adjusted by changing the number of Lagrangian particles, which is limited by device memory. Lagrangian computations scale with the number of nodes better than Eulerian computations because the former do not require collective communications. This means that the ratio of CPU and GPU computation times also depends on the number of nodes. Therefore, for a fixed number of Lagrangian particles, there is an optimal number of nodes, for which the time CPUs and GPUs work simultaneously is maximized. Scaling efficiency up to this optimal number of nodes is close to 100 %. Simulations that use both CPUs and GPUs take between 10 and 120 times less time and use between 10 to 60 times less energy than simulations run on CPUs only. Simulations with Lagrangian microphysics take up to 8 times longer to finish than simulations with Eulerian bulk microphysics, but the difference decreases as more nodes are used. The presented method of adaptation for computing clusters can be used in any numerical model with Lagrangian particles coupled to an Eulerian fluid flow.

The Role of the Unsteady Surface Wave‐Driven Ekman–Stokes Flow in the Accumulation of Floating Marine Litter

AbstractRecently, a number of authors have used global particle tracking simulations to identify the effect that different surface currents have on marine litter accumulation, including the role of surface waves through the Stokes drift. However, in the upper‐ocean boundary layer and in the presence of the Coriolis force, a wave‐driven Eulerian flow forms that must be superimposed onto the Stokes drift in order to obtain the correct Lagrangian velocity. Taking into account both the Coriolis–Stokes force and the surface wave stress, Higgins et al. (2020), https://doi.org/10.1029/2020GL089189 derived an expression for this unsteady wave‐driven Eulerian‐mean flow in the form of a convolution between the Stokes drift and the so‐called Ekman–Stokes kernel. In this paper, we apply this Ekman–Stokes kernel to generate a 12‐year global hindcast of the wave‐driven Eulerian current and show that its inclusion in particle tracking simulations has a significant effect on the distribution of small floating marine litter, such as microplastics. Using Lagrangian simulations, we find that the wave‐driven Eulerian current is sensitive to the value of viscosity but generally opposes the dispersive behavior of the Stokes drift, reducing the amount of cross‐Equator particle transport and transport to the polar regions, resulting in closer agreement between modeled and observed microplastic distributions.

Aerodynamic Prediction of Time Duration to Becoming Infected with Coronavirus in a Public Place

The COVID-19 pandemic has caused panic and chaos that modern society has never seen before. Despite their paramount importance, the transmission routes of coronavirus SARS-CoV-2 remain unclear and a point of contention between the various sectors. Recent studies strongly suggest that COVID-19 could be transmitted via air in inadequately ventilated environments. The present study investigates the possibility of the aerosol transmission of coronavirus SARS-CoV-2 and illustrates the associated environmental conditions. The main objective of the current work is to accurately predict the time duration of getting an infection while sharing an indoor space with a patient of COVID-19 or similar viruses. We conducted a 3D computational fluid dynamics (CFD)-based investigation of indoor airflow and the associated aerosol transport in a restaurant setting, where likely cases of airflow-induced infection of COVID-19 caused by asymptomatic individuals were reported in Guangzhou, China. The Eulerian–Eulerian flow model coupled with the k-Ɛ turbulence approach was employed to resolve complex indoor processes, including human respiration activities, such as breathing, speaking, and sneezing. The predicted results suggest that 10 minutes are enough to become infected with COVID-19 when sharing a Table with coronavirus patients. The results also showed that although changing the ventilation rate will improve the quality of air within closed spaces, it will not be enough to protect a person from COVID-19. This model may be suitable for future engineering analyses aimed at reshaping public spaces and indoor common areas to face the spread of aerosols and droplets that may contain pathogens.

On the role of continuous symmetries in the solution of the three-dimensional Euler fluid equations and related models.

We review and apply the continuous symmetry approach to find the solution of the three-dimensional Euler fluid equations in several instances of interest, via the construction of constants of motion and infinitesimal symmetries, without recourse to Noether’s theorem. We show that the vorticity field is a symmetry of the flow, so if the flow admits another symmetry then a Lie algebra of new symmetries can be constructed. For steady Euler flows this leads directly to the distinction of (non-)Beltrami flows: an example is given where the topology of the spatial manifold determines whether extra symmetries can be constructed. Next, we study the stagnation-point-type exact solution of the three-dimensional Euler fluid equations introduced by Gibbon et al. (Gibbon et al. 1999 Physica D 132, 497–510. (doi:10.1016/S0167-2789(99)00067-6)) along with a one-parameter generalization of it introduced by Mulungye et al. (Mulungye et al. 2015 J. Fluid Mech. 771, 468–502. (doi:10.1017/jfm.2015.194)). Applying the symmetry approach to these models allows for the explicit integration of the fields along pathlines, revealing a fine structure of blowup for the vorticity, its stretching rate and the back-to-labels map, depending on the value of the free parameter and on the initial conditions. Finally, we produce explicit blowup exponents and prefactors for a generic type of initial conditions.This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.

Geometric microcanonical theory of two-dimensional truncated Euler flows.

This paper presents a geometric microcanonical ensemble perspective on two-dimensional truncated Euler flows, which contain a finite number of (Fourier) modes and conserve energy and enstrophy. We explicitly perform phase space volume integrals over shells of constant energy and enstrophy. Two applications are considered. In the first part, we determine the average energy spectrum for highly condensed flow configurations and show that the result is consistent with Kraichnan's canonical ensemble description, despite the fact that no thermodynamic limit is invoked. In the second part, we compute the probability density for the largest-scale mode of a free-slip flow in a square, which displays reversals. We test the results against numerical simulations of a minimal model and find excellent agreement with the microcanonical theory, unlike the canonical theory, which fails to describe the bimodal statistics. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Delta shock as free piston in pressureless Euler flows

We establish the equivalence of free piston and delta shock, for the one-space-dimensional pressureless compressible Euler equations. The delta shock appearing in the singular Riemann problem is exactly the piston that may move freely forward or backward in a straight tube, driven by the pressureless Euler flows on two sides of it in the tube. This result not only helps to understand the physics of the somewhat mysterious delta shocks, but also provides a way to reduce the fluid-solid interaction problem, which consists of several initial-boundary value problems coupled with moving boundaries, to a simpler Cauchy problem. We show the equivalence from three different perspectives. The first one is from the sticky particles, and derives the ordinary differential equation (ODE) of the trajectory of the piston by a straightforward application of conservation law of momentum, which is physically simple and clear. The second one is to study a coupled initial-boundary value problem of pressureless Euler equations, with the piston as a moving boundary following the Newton's second law. It depends on a concept of Radon measure solutions of initial-boundary value problems of the compressible Euler equations which enables us to calculate the force on the piston given by the flow. The last one is to solve directly the singular Riemann problem and obtain the ODE of delta shock by the generalized Rankine-Hugoniot conditions. All the three methods lead to the same ODE.

Existence of stationary vortex sheets for the 2D incompressible Euler equation

AbstractWe construct a new type of planar Euler flows with localized vorticity. Let $\kappa _i\not =0$ , $i=1,\ldots , m$ , be m arbitrarily fixed constants. For any given nondegenerate critical point $\mathbf {x}_0=(x_{0,1},\ldots ,x_{0,m})$ of the Kirchhoff–Routh function defined on $\Omega ^m$ corresponding to $(\kappa _1,\ldots , \kappa _m)$ , we construct a family of stationary planar flows with vortex sheets that have large vorticity amplitude and concentrate on curves perturbed from small circles centered near $x_{0,i}$ , $i=1,\ldots ,m$ . The proof is accomplished via the implicit function theorem with suitable choice of function spaces.

Nonlinear stability of planar steady Euler flows associated with semistable solutions of elliptic problems

This paper is devoted to the study of nonlinear stability of steady incompressible Euler flows in two dimensions. We prove that a steady Euler flow is nonlinearly stable in L p L^p norm of the vorticity if its stream function is a semistable solution of some semilinear elliptic problem with nondecreasing nonlinearity. The idea of the proof is to show that such a flow has strict local maximum energy among flows whose vorticities are rearrangements of a given function, with the help of an improved version of Wolansky and Ghil’s stability theorem. The result can be regarded as an extension of Arnol’d’s second stability theorem.

Nonuniqueness and Existence of Continuous, Globally Dissipative Euler Flows

We show that Hölder continuous incompressible Euler flows that satisfy the local energy inequality (“globally dissipative” solutions) exhibit nonuniqueness and contain examples that strictly dissipate kinetic energy. The collection of such solutions emanating from a fixed initial data may have positive Hausdorff dimension in the energy space even if the local energy equality is imposed, and the set of initial data giving rise to such an infinite family of solutions is $$C^0$$ dense in the space of continuous, divergence free vector fields on the torus $${{\mathbb {T}}}^3$$ . The construction of these solutions involves a new and explicit convex integration approach mirroring Kraichnan’s LDIA theory of turbulent energy cascades that overcomes the limitations of previous schemes, which had been restricted to bounded measurable solutions or to continuous solutions that dissipate total kinetic energy.

Existence of a conjugate point in the incompressible Euler flow on an ellipsoid

Existence of a conjugate point in the incompressible Euler flow on a sphere and an ellipsoid is considered. Misiołek (1996) formulated a differential-geometric criterion (we call the M-criterion) for the existence of a conjugate point in a fluid flow. In this paper, it is shown that no zonal flow (stationary Euler flow) satisfies the M-criterion if the background manifold is a sphere, on the other hand, there are zonal flows satisfy the M-criterion if the background manifold is an ellipsoid (even it is sufficiently close to the sphere). The conjugate point is created by the fully nonlinear effect of the inviscid fluid flow with differential geometric mechanism.

Experimental and numerical investigations of solitary wave-induced non-equilibrium scour around structure of square cross-section on sandy berm

A series of laboratory experiments and high-fidelity numerical simulations were carried out to investigate the scour induced by a solitary wave around a non-slender, vertical structure of a square cross-section on a sandy berm. Various wave and water level combinations, structure layouts, and dimensions were studied. The flow separation at the sharp edges resulted in counterclockwise rotating out-of-plane vortices that extended from the free surface to the sandy berm. These energetic vortices were found to be the primary driving mechanism of the scouring through entraining, and entrapping the sand particles into their cores, keeping them in suspension, carrying them, and releasing them as the vortices propagate along a spiral trajectory. The analyses also showed that the flow blockage by the structure could result in flow field modulation and exacerbate the scour processes. The structure dimension appeared to be the most influential factor in the scouring process, drastically altering the characteristics of the non-equilibrium scour holes. Irrespective of the structure dimension or layout, the scour depth was greater around the seaside edge of the structure while the scour width was larger on the leeside edge. The uncertainties associated with the maximum non-equilibrium scour depth were quantified via Monte Carlo simulations which showed that the impact of the structure dimension on the maximum scour depth was almost twice more significant than that of the layout. The three-dimensional (3D) Eulerian two-phase flow numerical model, SedWaveFoam, was shown to be able to simulate sediment transport and scour around sharp-edged square structure at a low Keulegan-Carpenter ( K C ) number. • Interactions between a solitary wave and an impermeable structure placed on an erodible berm were studied experimentally. • Energetic vortices, formed at the corners of the structure, were found to be the primary driving mechanism of the scouring. • Structure dimension was the most influential factor in the scouring process. • Monte Carlo simulation was employed to assess the sensitivity of the maximum non-equilibrium scour depth to flow conditions. • SedWaveFoam model simulated scour around sharp-edged square structure at low Keulegan-Carpenter (KC) number.

Clustered travelling vortex rings to the axisymmetric three-dimensional incompressible Euler flows

For the three dimensional axisymmetric Euler flow, we construct a family of solutions with multiple travelling vortex rings, with large speed of order O(|lnɛ|), where ɛ>0 is a small parameter. Our construction is based on the analysis of the following nonlinear elliptic equation: ∂rrψ+3r∂rψ+∂zzψ=−F((ψ−α2|lnɛ|)r2),(r,z)∈R2,ψr(0,z)=0forr=0,for some special functions F, where α is a parameter. The location of the vortex rings is governed by some balancing systems, which can be solved by the polynomial method in several special cases. For the non-swirl case, in the core of each vortex ring, our solutions can be regarded as a rescaled finite mass solution of the Liouville equation. The results can be generalized directly to the case with swirl, for which we also construct different types of solutions with multiple vortex rings.

Insight into the sensitivities of freckles in the directional solidification of single-crystal turbine blades

Freckle, a thermal-solutal-fluid flow induced metallurgical defect, can degrade the mechanical performance of the investment cast nickel-based single-crystal turbine aerofoil. Channel segregation, appearing as channel-liked solute enrichment pattern, is generally considered to be a representative phenomenon in the formation of freckles. The geometrical effect of the intricate curve turbine blade with relevant processing conditions into the freckle mechanism is still unprecedented. In this work, we proposed the Eulerian multiphase thermal-fluid flow model for predicting the freckles (channel segregation) in the representative single crystal turbine blades. The prediction which captures the complex transport phenomena of thermal-solutal convection, solute plumes and segregation profile is in good agreement with the available experimentation. Based upon the thermal-solutal-fluid flow rationalisation of the solidification process, mechanisms proposed provide better understanding of the channel segregation sensitivities to the cooling conditions and geometrical factors. Sensitivity studies verify the mechanisms and give further implication on preventing and even eliminating the freckles.

On the singular Weinstein conjecture and the existence of escape orbits for b-Beltrami fields

Motivated by Poincaré’s orbits going to infinity in the (restricted) three-body problem [see H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Vol. 3 (Gauthier-Villars, 1899) and A. Chenciner, Poincaré and the three-body problem, in Henri Poincaré, 1912–2012 (Birkhäuser, Basel, 2015), pp. 51–149], we investigate the generic existence of heteroclinic-like orbits in a neighborhood of the critical set of a [Formula: see text]-contact form. This is done by using a singular counterpart [R. Cardona, E. Miranda and D. Peralta-Salas, Euler flows and singular geometric structures, Philos. Trans. R. Soc. A 377(2158) (2019) 20190034] of Etnyre–Ghrist’s contact/Beltrami correspondence [J. Etnyre and R. Ghrist, Contact topology and hydrodynamics: I. Beltrami fields and the Seifert conjecture, Nonlinearity 13(2) (2000) 441–458], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [Generic properties of eigenfunctions, Amer. J. Math. 98(4) (1976) 1059–1078]. Specifically, we analyze the [Formula: see text]-Beltrami vector fields on [Formula: see text]-manifolds of dimension [Formula: see text] and prove that for a generic asymptotically exact [Formula: see text]-metric they exhibit escape orbits. We also show that a generic asymptotically symmetric [Formula: see text]-Beltrami vector field on an asymptotically flat [Formula: see text]-manifold has a generalized singular periodic orbit and at least four escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose [Formula: see text]- and [Formula: see text]-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture.

Rough solutions of the 3-D compressible Euler equations

We prove the local-in-time well-posedness for the solution of the compressible Euler equations in 3-D for the Cauchy data of the velocity, density and vorticity $(v,\varrho\mathfrak{w} \in H^s \times H^s \times H^{s'}$, $2\lt s'\lt s$. The result extends the sharp results of Smith-Tataru and of Wang, established in the irrotational case, i.e., $\mathfrak{w}=0$, which is known to be optimal for $s>2$. At the opposite extreme, in the incompressible case, i.e., with a constant density, the result is known to hold for $\mathfrak{w}\in H^s$, $s>3/2$ and fails for $s\le 3/2$. We therefore, conjecture that the optimal result should be $(v,\varrho,\mathfrak{w} \in H^s\times H^s\times H^{s'}$, $s>2$, $s' > \frac{3}{2}$. We view our work here as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of sound wave from the transported part, and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustical spacetime.

Numerical Simulation of Conical and Linear-Shaped Charges Using an Eulerian Elasto-Plastic Multi-Material Multi-Phase Flow Model with Detonation.

This study developed a hydrocode to numerically simulate both conical and linear-shaped charges using an Eulerian multi-material and multi-phase flow model. Elasto-plastic solids and the detonation of a high explosive charge were modeled using a Johnson–Cook material model and the programmed burn model, respectively. Further, the plasticity of the solids was calculated using a radial return mapping algorithm. The model was solved using a high-resolution computational fluid dynamics (CFD) technique on Cartesian grids. Material interfaces were tracked using the level-set method, and the boundary conditions were imposed using the ghost fluid method. The developed hydrocode was validated using high-speed impact problems. Consequently, the developed hydrocode was used to successfully simulate the evolution and penetration of metal jets in shaped charges after a detonation.