Inviscid Euler Equations Research Articles

Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on T×R. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L 2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically driven to small scales by a linear evolution and weakly converges as t→±∞. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. This convergence was formally derived at the linear level by Kelvin in 1887 and it occurs at an algebraic rate first computed by Orr in 1907; our work appears to be the first rigorous confirmation of this behavior on the nonlinear level.

Implementing a discontinuous Galerkin method for the compressible, inviscid Euler equations in the DUNE framework

AbstractA discontinuous Galerkin scheme was implemented in the DUNE framework to solve the compressible, inviscid Euler equations in a multi‐dimensional Cartesian grid. It uses a HLLC Riemann solver for the numerical fluxes in the interfaces, a total variation bounded limiter to handle discontinuities, a positivity preserving limiter for near vacuum conditions, and adaptive mesh refinement (AMR). (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Multidisciplinary pre-design of supersonic aircraft

Forecasts predict a strong market for supersonic travel in the business sector. It is especially appealing for high-net-worth individuals because of both, the reduction of travel time and prestige. The ecological and economic challenges that are related to supersonic flight need adequate answers in terms of technology and tools to evaluate aircraft for the described task. This paper gives a short overview on research that has been conducted in the past, followed by the presentation of the work done by the authors. The integration of methods for the evaluation of supersonic business jets in the context of aircraft conceptual design is outlined. The determination of aerodynamic coefficients and loads is done by solving the inviscid Euler equations. Furthermore the propulsion module is extended and a mission analysis tool, which is used for fuel mass estimation is briefly explained. The methods for structural analysis are detailed. At the end the verification based on a reference configuration is described.

Hermite WENO limiting for multi-moment finite-volume methods using the ADER-DT time discretization for 1-D systems of conservation laws

New Hermite Weighted Essentially Non-Oscillatory (HWENO) interpolants are developed and investigated within the Multi-Moment Finite-Volume (MMFV) formulation using the ADER-DT time discretization. Whereas traditional WENO methods interpolate pointwise, function-based WENO methods explicitly form a non-oscillatory, high-order polynomial over the cell in question. This study chooses a function-based approach and details how fast convergence to optimal weights for smooth flow is ensured. Methods of sixth-, eighth-, and tenth-order accuracy are developed. These are compared against traditional single-moment WENO methods of fifth-, seventh-, ninth-, and eleventh-order accuracy to compare against more familiar methods from literature. The new HWENO methods improve upon existing HWENO methods (1) by giving a better resolution of unreinforced contact discontinuities and (2) by only needing a single HWENO polynomial to update both the cell mean value and cell mean derivative.Test cases to validate and assess these methods include 1-D linear transport, the 1-D inviscid Burger's equation, and the 1-D inviscid Euler equations. Smooth and non-smooth flows are used for evaluation. These HWENO methods performed better than comparable literature-standard WENO methods for all regimes of discontinuity and smoothness in all tests herein. They exhibit improved optimal accuracy due to the use of derivatives, and they collapse to solutions similar to typical WENO methods when limiting is required. The study concludes that the new HWENO methods are robust and effective when used in the ADER-DT MMFV framework. These results are intended to demonstrate capability rather than exhaust all possible implementations.

Evolution of cellular structures on oblique detonation surfaces

In this study, numerical simulations using the inviscid Euler equations with one-step Arrhenius chemistry model are carried out to investigate the effects of activation energy and wedge angle on the stability of oblique detonation surfaces. Two kinds of cellular structure are studied, one is featured by a single group of transverse waves traveling upstream, referred to as LRTW (left-running transverse waves), and the other is featured by additional RRTW (right-running transverse waves). The present computational simulation reveals the formation of un-reacted gas pockets behind the cellular oblique detonation. Numerical smoked foil records are produced to show the emergence of the two types of transverse waves and the evolution of the unstable cellular structure of the oblique detonation. The transverse wave dynamics, including the colliding, emerging and splitting types, are found to be similar to the normal detonation propagation, demonstrating the instability mechanism is originated from the inherent instability of cellular detonations. Statistical analysis on the cellular structure is carried out to observe quantitatively the influences of activation energy and wedge angle. Results from the parametric study show that high activation energy and low wedge angle are favorable to the LRTW formation. However, the condition for the RRTW formation is more complex. In the case of low activation energy, small wedge angle is beneficial to the RRTW formation, as to the LRTW formation. In contrary, for high activation energy, there appears one moderate wedge angle favoring the RRTW formation and giving the shortest length between the onset of both LR and RR transverse waves. For quantitative comparison, we analyze the variation of two distances with the wedge angle, one is between the detonation initiation and LRTW formation points, and the other between LRTW and RRTW formation points. Results show the latter is relatively less pronounced than the former, indicating the RRTW formation depends mainly on the activation energy and the generation of LRTW.

Rotor aerodynamic power limits at low tip speed ratio using CFD

When investigating limits of rotor aerodynamic models, the Betz limit serves as a solid marker of an upper limit which no model should be able to exceed. A century ago Joukowsky (1912) proposed a rotor aerodynamic model utilizing a rotating actuator disc with a constant circulation. This model has since then been the subject of much controversy as it predicts a power performance that for all tip speed ratios exceeds the Betz limit and which goes to infinity when the tip speed ratio goes to zero. Recently, it was demonstrated that the Joukowsky model is fully consistent with the inviscid Euler equations and that the apparent inconsistency partly can be explained by the lack of viscous effects (Sprensen and van Kuik [4]). However, even including a term to account for the effect of viscosity at small tip speed ratios, the model still predicts a power yield that exceeds the Betz limit. In the present work we study in detail, using a CFD actuator line model, the flow behavior for rotors at small tip speed ratios. It is shown that the excessive swirl appearing towards the rotor center at small tip speed ratios generates vortex breakdown, causing a recirculating zone in the wake that limits the power yield of the rotor. The appearance of vortex breakdown has a similar effect on the flow behavior as the vortex ring state that usually appears at higher tip speed ratios. Limits to where vortex breakdown might occur with tip speed ratio and rotor loading as parameter are investigated and presented in the paper. The limits found correspond to well-known criterion for vortex breakdown onset for swirling flows in general. By applying a criterion for vortex breakdown in combination with the general momentum theory, the power performance always stays below the Betz limit.

Waves on a Stuart vortex

We consider the linear stability of Stuart vortices, an exact periodic solution of the two-dimensional inviscid Euler equations of hydrodynamics consisting of an infinite row of co-rotating vortices, to wavelike disturbances when the vortex amplitude is small but still much larger than the disturbances. An unsteady critical layer analysis is used, with outer and inner expansions away from and near the vortex cores which we match to obtain disturbance growth rates. Disturbance-vortex interactions occur inside the critical layer and are neglected elsewhere. We consider α+-1∼O(1) and α+≈1 separately, where α+ is ratio of the disturbance wavenumber to that of the periodic vortex row. We consider primarily three-dimensional (oblique) disturbances but touch upon two-dimensional disturbances in the vortex plane. There are resonance mechanisms for oblique disturbances when k+⩽1, with k+ the ratio of the streamwise disturbance wavenumber to that of the periodic vortex row. When k+=1, this is fundamental mode or Benney–Lin instability, thought to be connected to streamwise streaks seen in experiments, and when k+=1/2, this is subharmonic instability or helical pairing, thought to be connected to vortex pairing.

A comparison of computational efficiencies of spectral difference method and correction procedure via reconstruction

We report computational efficiencies of two types of numerical solvers. The first type uses the spectral difference (SD) method and the second one uses the correction procedure via reconstruction (CPR). In this paper, we employ the lumped g2 scheme proposed by Huynh as an example of the CPR approach. The solvers deal with both inviscid Euler equations and Navier–Stokes equations on 2D unstructured grids which are comprised of all quadrilateral cells. Both types of solvers are programmed using Fortran 90 with similar management of data structures. We employ identical time marching schemes for both SD and CPR methods. Spatial 3rd and 4th order of accuracy for both methods is demonstrated by a study of a moving inviscid vortex. The comparisons were directed to measure the computational efficiency of both SD and CPR methods in spatial discretization. With respect to the fourth order methods, CPR is 27% faster than SD for inviscid flow, and more promisingly, CPR is over 40% faster than SD for viscous flow.

A logarithmically improved blow-up criterion of smooth solutions for nematic liquid crystal flows with partial viscosity

In this paper we investigate the Cauchy problem for nematic liquid crystal flows with partial viscosity in R 3 . A logarithmically improved blow-up criterion of smooth solutions is established. The result is analogous to the celebrated Beale-Kato-Majda type criterion for the inviscid Euler equations of incompressible fluids.

Hysteresis phenomenon at interaction of shock waves generated by a cylinder array

This paper describes investigations of the interaction between bow shock waves generated by cylindrical bodies in a supersonic flow. Numerical simulations are performed using the inviscid Euler equations for cylinders whose axes are parallel to each other and normal to the flow direction. Mostly an infinite periodical cylinder array is considered, but the case of two cylinders is also briefly discussed. Three different regimes of the shock wave interaction, a regular interaction, a Mach interaction, and a choked flow, have been observed for the flow through the periodical cylinder array. In the case of the flow around two bodies, the choked flow is replaced by a regime with a collective bow shock. The transition between different flow regimes is studied by varying the inflow Mach number or the distance between the cylinders. A hysteretic behavior at the transition between the regular and Mach interactions has been observed. The transition is governed by the theoretical detachment and von Neumann criteria based on the local shock wave inclination at the interaction point.

Viscous limit to contact discontinuity for the 1-D compressible Navier–Stokes equations

In this paper, we study the zero dissipation limit problem for the one-dimensional compressible Navier–Stokes equations. We prove that if the solution of the inviscid Euler equations is piecewise constants with a contact discontinuity, then there exist smooth solutions to the Navier–Stokes equations which converge to the inviscid solution away from the contact discontinuity at a rate of κ 3 4 as the heat-conductivity coefficient κ tends to zero, provided that the viscosity μ is higher order than the heat-conductivity κ or the same order as κ. Here we have no need to restrict the strength of the contact discontinuity to be small.

Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations

The generalized Riemann problem (GRP) scheme for the Euler equations and gas-kinetic scheme (GKS) for the Boltzmann equation are two high resolution shock capturing schemes for fluid simulations. The difference is that one is based on the characteristics of the inviscid Euler equations and their wave interactions, and the other is based on the particle transport and collisions. The similarity between them is that both methods can use identical MUSCL-type initial reconstructions around a cell interface, and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time-dependent flux function. Although both methods have been applied successfully to the inviscid compressible flow computations, their performances have never been compared. Since both methods use the same initial reconstruction, any difference is solely coming from different underlying mechanism in their flux evaluation. Therefore, such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances. Since GRP is so faithfully solving the inviscid Euler equations, the comparison can be also used to show the validity of solving the Euler equations itself. The numerical comparison shows that the GRP exhibits a slightly better computational efficiency, and has comparable accuracy with GKS for the Euler solutions in 1D case, but the GKS is more robust than GRP. For the 2D high Mach number flow simulations, the GKS is absent from the shock instability and converges to the steady state solutions faster than the GRP. The GRP has carbuncle phenomena, likes a cloud hanging over exact Riemann solvers. The GRP and GKS use different physical processes to describe the flow motion starting from a discontinuity. One is based on the assumption of equilibrium state with infinite number of particle collisions, and the other starts from the non-equilibrium free transport process to evolve into an equilibrium one through particle collisions. The different mechanism in the flux evaluation deviates their numerical performance. Through this study, we may conclude scientifically that it may NOT be valid to use the Euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution. To adapt the Navier–Stokes (NS) equations is NOT valid either because the NS equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity. This fact alludes to the consistency of the Euler and Navier–Stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high Mach number flows. The development of numerical algorithm is similar to the modeling process in deriving the governing equations, but the control volume here cannot be shrunk to zero.

A Beale-Kato-Madja Criterion for Magneto-Micropolar Fluid Equations with Partial Viscosity

We study the incompressible magneto-micropolar fluid equations with partial viscosity in . A blow-up criterion of smooth solutions is obtained. The result is analogous to the celebrated Beale-Kato-Majda type criterion for the inviscid Euler equations of incompressible fluids.

Blade Excitation in Pulse-Charged Mixed-Flow Turbocharger Turbines

In this article, a fully three-dimensional computational modeling approach in the time and frequency domain is presented, which allows to accurately predicting fluid-structure interactions in pulse-charged mixed-flow turbocharger turbines. As part of the approach, a transient computational fluid mechanics analysis is performed based on the compressible inviscid Euler equations covering an entire engine cycle. The resulting harmonic orders of aerodynamic excitation are imposed in a forced response analysis of the respective eigenvector to determine effective stress amplitudes. The modeling approach is validated with experimental results based on various mixed-flow turbine designs. It is shown that the numerical results accurately predict the measured stress levels. The numerical approach can be used in the turbine design and optimization process. Aerodynamic excitation forces are the main reason for high cycle fatigue in turbocharger turbines and therefore a fundamental understanding is of key importance.

Reduced-Order Model for Viscous Aerodynamic Flow Past an Airfoil

A method is presented to obtain reduced-order models of multiparametric, steady, transonic aerodynamic flows around two-dimensional airfoils. The method is based on proper orthogonal decomposition and appropriate projection on the proper orthogonal decomposition manifold of the governing equations and boundary conditions, and it includes various ingredients that are new in this field. Proper orthogonal decomposition modes are obtained from a set of snapshots calculated by computational fluid dynamics based on the compressible Navier―Stokes equations and a turbulence model, but projection on the proper orthogonal decomposition manifold is made using the inviscid Euler equations, which makes the method independent of the computational fluid dynamics scheme; projection is made using only a limited number of mesh points, obtaining quite good results. Shock waves are treated specifically in the proper orthogonal decomposition description, to avoid the need of using a too-large number of snapshots. The method is checked and discussed on a specific aerodynamic problem.

A high-order gas-kinetic Navier–Stokes flow solver

The foundation for the development of modern compressible flow solver is based on the Riemann solution of the inviscid Euler equations. The high-order schemes are basically related to high-order spatial interpolation or reconstruction. In order to overcome the low-order wave interaction mechanism due to the Riemann solution, the temporal accuracy of the scheme can be improved through the Runge–Kutta method, where the dynamic deficiencies in the first-order Riemann solution is alleviated through the sub-step spatial reconstruction in the Runge–Kutta process. The close coupling between the spatial and temporal evolution in the original nonlinear governing equations seems weakened due to its spatial and temporal decoupling. Many recently developed high-order methods require a Navier–Stokes flux function under piece-wise discontinuous high-order initial reconstruction. However, the piece-wise discontinuous initial data and the hyperbolic-parabolic nature of the Navier–Stokes equations seem inconsistent mathematically, such as the divergence of the viscous and heat conducting terms due to initial discontinuity. In this paper, based on the Boltzmann equation, we are going to present a time-dependent flux function from a high-order discontinuous reconstruction. The theoretical basis for such an approach is due to the fact that the Boltzmann equation has no specific requirement on the smoothness of the initial data and the kinetic equation has the mechanism to construct a dissipative wave structure starting from an initially discontinuous flow condition on a time scale being larger than the particle collision time. The current high-order flux evaluation method is an extension of the second-order gas-kinetic BGK scheme for the Navier–Stokes equations (BGK-NS). The novelty for the easy extension from a second-order to a higher order is due to the simple particle transport and collision mechanism on the microscopic level. This paper will present a hierarchy to construct such a high-order method. The necessity to couple spatial and temporal evolution nonlinearly in the flux evaluation can be clearly observed through the numerical performance of the scheme for the viscous flow computations.

Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier–Stokes equations

In this paper, we study the zero dissipation limit problem for the one-dimensional compressible Navier–Stokes equations. We prove that if the solution of the inviscid Euler equations is piecewise constants with a contact discontinuity, then there exist smooth solutions to the Navier–Stokes equations which converge to the inviscid solution away from the contact discontinuity at a rate of κ 1 4 as the heat-conductivity coefficient κ tends to zero, provided that the viscosity μ is of higher order than the heat-conductivity κ. Without loss of generality, we set μ ≡ 0 . Here we have no need to restrict the strength of the contact discontinuity to be small.

Resolution of d’Alembert’s Paradox

We propose a resolution of d’Alembert’s Paradox comparing observation of substantial drag/lift in fluids with very small viscosity such as air and water, with the mathematical prediction of zero drag/lift of stationary irrotational solutions of the incompressible inviscid Euler equations, referred to as potential flow. We present analytical and computational evidence that (i) potential flow cannot be observed because it is illposed or unstable to perturbations, (ii) computed viscosity solutions of the Euler equations with slip boundary conditions initiated as potential flow, develop into turbulent solutions which are wellposed with respect to drag/lift and which show substantial drag/lift, in accordance with observations.

HyPAM: A hybrid continuum-particle model for incompressible free-surface flows

Three major issues associated with numerical simulations of complex free-surface flows, viz. interface tracking, fragmentation and large physical jumps, are addressed by a new hybrid continuum-particle model (HyPAM). The new model consists of three parts: (1) the Polygonal Area Mapping method [Q. Zhang, P.L.-F. Liu, A new interface tracking method: the polygonal area mapping method, J. Comput. Phys. 227(8) (2008) 4063-4088]; (2) a new algorithm that decomposes the interested (water) phase into a continuum zone, a buffer zone and a particle zone, based on material topology and graph theory; (3) a ‘passive-response’ assumption, in which the air phase is assumed to respond passively to the continuum part of the water phase. The incompressible inviscid Euler equations and the equations describing the free fall of rigid bodies are used as the governing equations for the continuum-buffer zone and the particle zone, respectively, and separately. A number of examples, including water droplet impact, solitary wave propagation, and dam-break problems, are simulated for the illustration and validation of HyPAM. It is shown that HyPAM is more accurate and versatile than a continuum-based Volume-of-Fluid model. One major contribution of this work is the single-phase decomposition algorithm, useful for many other hybrid formulations. Neglecting surface tension, viscosity and particle interactions, HyPAM is currently limited to mildly-fragmented free-surface flows with high Reynolds and Weber numbers.

Sub‐cell shock capturing and spacetime discontinuity tracking for nonlinear conservation laws

AbstractWe describe two approaches to resolving shocks and other discontinuities in spacetime discontinuous Galerkin (SDG) methods for nonlinear conservation laws. The first is an adaptation of the sub‐cell shock‐capturing technique, recently introduced by Persson and Peraire, to the special circumstances of SDG solutions constructed on causal spacetime grids. We restrict the stabilization operator to spacetime element interiors, thereby preserving the 𝒪(N) computational complexity and the element‐wise conservation properties of the basic SDG method, and use a special discontinuity indicator to limit the stabilization to elements traversed by discontinuous solution features. The method resolves discontinuities within individual spacetime elements having a sufficiently high‐order basis. Numerical studies demonstrate the combination of sub‐cell shock capturing with h‐adaptive spacetime meshing that circumvents the projection errors inherent to purely spatial remeshing procedures.In a second method, we use adaptive spacetime meshing operations to track the trajectories of singular surfaces while maintaining the quality of the surrounding mesh. We present a series of feasibility studies where we track shocks and contact discontinuities in solutions to the inviscid Euler equations, including an example where the trajectories are not known a priori. The SDG‐tracking method sharply resolves discontinuities without mesh refinement and requires very little stabilization. Copyright © 2008 John Wiley & Sons, Ltd.