Compressible Inviscid Research Articles

Numerical 3D flow simulation of ultrasonic horns with attached cavitation structures and assessment of flow aggressiveness and cavitation erosion sensitive wall zones

As a contribution to a better understanding of cavitation erosion mechanisms, a compressible inviscid finite volume flow solver with barotropic homogeneous liquid–vapor mixture cavitation model is applied to ultrasonic horn set-ups with and without stationary specimen, that exhibit attached cavitation at the horn tip. Void collapses and shock waves, which are closely related to cavitation erosion, are resolved. The computational results are compared to hydrophone, shadowgraphy and erosion test data. At the horn tip, vapor volume and topology, subharmonic oscillation frequency as well as the amplitude of propagating pressure waves are in good agreement with experimental data. For the evaluation of flow aggressiveness and the assessment of erosion sensitive wall zones, statistical analyses of wall loads and of the multiplicity of distinct collapses in wall-adjacent flow regions are applied to the horn tip and the stationary specimen. An a posteriori projection of load collectives, i.e. cumulative collapse rate vs. collapse pressure, onto a reference grid eliminates the grid dependency effectively for attached cavitation at the horn tip, whereas a significant grid dependency remains at the stationary specimen. The load collectives show an exponential decrease towards higher collapse pressures. Erosion sensitive wall zones are well predicted for both, horn tip and stationary specimen, and load profiles are in good qualitative agreement with measured topography profiles of eroded duplex stainless steel samples after long-term runs. For the considered amplitude and gap width according to ASTM G32-10 standard, the analysis of load collectives reveals that the distinctive erosive ring shape at the horn tip can be attributed to frequent breakdown and re-development of a small portion of the tip-attached cavity. This partial breakdown of the attached cavity repeats at each driving cycle and is associated with relatively moderate collapse peak pressures, whereas the stationary specimen is rather unfrequently stressed at the end of each subharmonic oscillation cycle by the violent collapse of the complete cavity.

Elastic rocking vibration of an offshore Gravity Base Foundation

Gravity Base Foundation is one of the main foundations in offshore wind turbine structure, and the foundation rocking stiffness mostly controls the structure fatigue life. This study models the rocking vibration of a rigid disc foundation on a poro-elastic seabed, covered by a compressible inviscid seawater half-space. The coupled rocking problem is solved using the 3-dimensional elastic wave theory. The behavior of seawater and soil are depicted with the Euler equations and the Biot's theory, respectively. The contact surface of the disc foundation with the soil and the water is assumed to be smooth. The solutions are expressed in Fredholm integral equations of the second kind and solved numerically. The obtained dynamic impedances are used to study the dynamic behavior of Gravity Base Foundations of offshore wind turbines, which are very sensitive to dynamic moments. It is found that foundation with a permeable surface has much smaller dynamic amplitudes, compared to the one with an impermeable surface. The existence of seawater half-space will also alleviate the dynamic amplitudes, but decreasing the natural frequency of the foundation.

Hierarchical multi-dimensional limiting strategy for correction procedure via reconstruction

Hierarchical multi-dimensional limiting process (MLP) is improved and extended for flux reconstruction or correction procedure via reconstruction (FR/CPR) on unstructured grids. MLP was originally developed in finite volume method (FVM) and it provides an accurate, robust and efficient oscillation-control mechanism in multiple dimensions for linear reconstruction. This limiting philosophy can be hierarchically extended into higher-order Pn approximation or reconstruction. The resulting algorithm is referred to as the hierarchical MLP and facilitates detailed capture of flow structures while maintaining formal order-of-accuracy in a smooth region and providing accurate non-oscillatory solutions across a discontinuous region. This algorithm was developed within modal DG framework, but it can also be formulated into a nodal framework, most notably the FR/CPR framework. Troubled-cells are detected by applying the MLP concept, and the final accuracy is determined by a projection procedure and the hierarchical MLP limiting step. Extensive numerical analyses and computations, ranging from two-dimensional to three-dimensional fluid systems, have demonstrated that the proposed limiting approach yields outstanding performances in capturing compressible inviscid and viscous flow features.

Numerical 3D flow simulation of attached cavitation structures at ultrasonic horn tips and statistical evaluation of flow aggressiveness via load collectives

A compressible inviscid flow solver with barotropic cavitation model is applied to two different ultrasonic horn set-ups and compared to hydrophone, shadowgraphy as well as erosion test data. The statistical analysis of single collapse events in wall-adjacent flow regions allows the determination of the flow aggressiveness via load collectives (cumulative event rate vs collapse pressure), which show an exponential decrease in agreement to studies on hydrodynamic cavitation [1]. A post-processing projection of event rate and collapse pressure on a reference grid reduces the grid dependency significantly. In order to evaluate the erosion-sensitive areas a statistical analysis of transient wall loads is utilised. Predicted erosion sensitive areas as well as temporal pressure and vapour volume evolution are in good agreement to the experimental data.

Adjoint-based adaptive finite element method for the compressible Euler equations using finite calculus

In this paper, an adjoint-based error estimation and mesh adaptation framework is developed for the compressible inviscid flows. The algorithm employs the Finite Calculus (FIC) scheme for the numerical solution of the flow and discrete adjoint equations in the context of the Galerkin finite element method (FEM) on triangular grids. The FIC scheme treats the instabilities normally generated in the numerical solution of the fluid equations through adding two stabilization terms, called streamline term and transverse term, to the original central-based discretized formulation. The non-linear system of equations resulting from the flow problem is solved implicitly using a damped Newton's method accompanied with the exact Jacobian matrix. A defect corrected scheme is implemented to iteratively solve the linear system of equations related to the adjoint problem benefiting from the transpose of the Jacobian matrix. At each iteration, the linear systems of equations resulting from the fluid and adjoint problems are solved using a preconditioned GMRES method. Having calculated the error of a specified output functional locally, an h-refinement methodology based on the element subdivision is performed to refine the candidate elements. The quality of the numerical results proves the capability of the presented approach for the adjoint-based error estimation and mesh adaptation problems in different flow regimes.

Stability of compressible streamwise vortices

A sufficient condition for instability of full compressible inviscid streamwise vortices has been derived using asymptotic analysis with large wavenumbers. Based on the nature of instability, the obtained growth rate is described using an entropy instability term, a spiral instability term, a helicity instability term, and an acoustic instability term. The entropy instability term correlates with compressibility. The spiral instability term is reasonably expressed by the axial and azimuthal velocities and their wavenumbers. The helicity instability term depends on the helicity profiles. In particular, the unstable condition for wake-type flows is satisfied when they possess a negative helicity in the profiles. The acoustic instability term depends on the local speed of sound. Linear and nonlinear numerical simulations are used to validate the instability conditions. The effect of the helicity instability is shown to be significantly strong by the simulations.

Equivalence of a compressible inviscid flow and the Bloch vector under the thermal Jaynes–Cummings model

In this paper, we show that the time evolution of the Bloch vector governed by the thermal Jaynes–Cummings model is equivalent to a compressible inviscid flow with zero vorticity. Because of its quasiperiodicity, the dynamics of the Bloch vector includes countably infinite angular momenta as integrals of motion. Moreover, to derive the Bloch vector, we trace out the Hilbert space of the cavity field and remove entanglement between the single atom and the cavity mode. These facts indicate that the dynamics of the Bloch vector can be described with a hidden-variable model that has local determinism and a countably infinite number of degrees of freedom. Our results fit these considerations.

Solution of Turbine Blade Cascade Flow Using an Improved Panel Method

An improved panel method has been developed to calculate compressible inviscid flow through a turbine blade row. The method is a combination of the panel method for infinite cascade, a deviation angle model, and a compressibility correction. The resulting solution provides a fast flexible mesh-free calculation for cascade flow. A VKI turbine blade cascade is used to evaluate the method, and the comparison with experiment data is presented.

Global weak solutions to 1D compressible Euler equations with radiation

Global weak solutions to the 1D compressible Euler equations with radiation SUMMARY We consider the Cauchy problem for the equations of one-dimensional motion of a compressible inviscid gas coupled with radiation through a radiative transfer equation. Assuming suitable hypotheses on the transport coefficients and the data, we prove that the problem admits a weak solution. More precisely, we show that a sequence of approximate solutions constructed by a generalized Glimm's scheme admits a subsequence converging to an entropic solution of the problem.

Numerical Solution of Inviscid Compressible Steady Flows around the RAE 2822 Airfoil

The article presents results of a numerical solution of subsonic, transonic and supersonic flows described by the system of Euler equations in 2D compressible flows around the RAE 2822 airfoil. Authors used FVM multistage Runge-Kutta method to numerically solve the flows around the RAE 2822 airfoil. The results are compared with the solution using the software Ansys Fluent 15.0.7.

Construction of energy-stable projection-based reduced order models

An approach for building energy-stable Galerkin reduced order models (ROMs) for linear hyperbolic or incompletely parabolic systems of partial differential equations (PDEs) using continuous projection is developed. This method is an extension of earlier work by the authors specific to the equations of linearized compressible inviscid flow. The key idea is to apply to the PDEs a transformation induced by the Lyapunov function for the system, and to build the ROM in the transformed variables. For linear problems, the desired transformation is induced by a special inner product, termed the “symmetry inner product”, which is derived herein for several systems of physical interest. Connections are established between the proposed approach and other stability-preserving model reduction methods, giving the paper a review flavor. More specifically, it is shown that a discrete counterpart of this inner product is a weighted L2 inner product obtained by solving a Lyapunov equation, first proposed by Rowley et al. and termed herein the “Lyapunov inner product”. Comparisons between the symmetry inner product and the Lyapunov inner product are made, and the performance of ROMs constructed using these inner products is evaluated on several benchmark test cases.

A Framework for Curved Boundary Representation in 2D Discontinuous Galerkin Euler Solvers

Finite-element based CFD solvers like the family of Discontinuous Galerkin (DG) solvers suffer severely from inaccurate boundary reconstruction. In this matter, developing an accurate and flexible strategy is highly demanded to provide high-order curved boundary representation in DG simulations. In this paper, a general framework is introduced to design the curved elements in discontinuous Galerkin finite-element (DGFEM) simulations. The aim is to connect the boundary to the surrounding mesh by defining an appropriate set of basis functions which deliver the curvature information inside the mesh region adjacent to the boundary. This information is then used in flux integral calculations. The proposed framework is applied in Lagragian and Hermitian boundary representations. The efficiency of the method is analyzed for compressible inviscid flow test cases using the discontinuous Galerkin scheme. It is illustrated that using the curved-side elements in the present approach, is adequate to reduce the artificial entropy generation near the boundaries. This leads to the simulations with the desired order of accuracy. The results show a well consistency in h/p-refinement which advocates the use of the proposed approach in high-order CFD simulations.

A relaxation Riemann solver for compressible two-phase flow with phase transition and surface tension

The dynamics of two-phase flows depend crucially on interfacial effects like surface tension and phase transition. A numerical method for compressible inviscid flows is proposed that accounts in particular for these two effects. The approach relies on the solution of Riemann-like problems across the interface that separates the liquid and the vapour phase. Since the analytical solutions of the Riemann problems are only known in particular cases an approximative Riemann solver for arbitrary settings is constructed. The approximative solutions rely on the relaxation technique.The local well-posedness of the approximative solver is proven. Finally we present numerical experiments for radially symmetric configurations that underline the reliability and efficiency of the numerical scheme.

A frame-invariant vector limiter for flux corrected nodal remap in arbitrary Lagrangian–Eulerian flow computations

This article describes a frame-invariant vector limiter for Flux-Corrected Transport (FCT) numerical methods. Our approach relies on an objective vector projection, and, because of its intrinsic structure, the proposed approach can be generalized with ease to higher-order tensor fields.The proposed concept is applied to nodal finite element formulations and the so-called algebraic FCT paradigm, but the ideas pursued here are very general and also apply to more general instantiations of flux-corrected transport.Specifically, we consider the arbitrary Lagrangian–Eulerian (ALE) equations of compressible inviscid flows. In addition to the geometric conservation law (GCL) and the local extreme diminishing (LED) property of the original scalar limiters, the proposed approach ensures frame invariance (objectivity) for vectors. Particularly, we use an ALE strategy based on a two-stage, Lagrangian plus mesh remap (data transfer based on conservative interpolation), in which remap and limiting are performed in a synchronized way. The proposed approach is however of general applicability, is not limited to a specific ALE implementation, and can easily be generalized to computations with standard (monolithic) ALE or Eulerian reference frames.The significance of the frame-invariant limiter for vectors is demonstrated in computations of compressible materials under extreme load conditions. Extensive testing in two and three dimensions demonstrates that the proposed limiter greatly enhances the robustness and reliability of the existing methods under typical computational scenarios.

A discontinuous Galerkin method with block cyclic reduction solver for simulating compressible flows on GPUs

An optimized implementation of a block tridiagonal solver based on the block cyclic reduction (BCR) algorithm is introduced and its portability to graphics processing units (GPUs) is explored. The computations are performed on the NVIDIA GTX480 GPU. The results are compared with those obtained on a single core of Intel Core i7-920 (2.67 GHz) in terms of calculation runtime. The BCR linear solver achieves the maximum speedup of 5.84x with block size of 32 over the CPU Thomas algorithm in double precision. The proposed BCR solver is applied to discontinuous Galerkin (DG) simulations on structured grids via alternating direction implicit (ADI) scheme. The GPU performance of the entire computational fluid dynamics (CFD) code is studied for different compressible inviscid flow test cases. For a general mesh with quadrilateral elements, the ADI-DG solver achieves the maximum total speedup of 7.45x for the piecewise quadratic solution over the CPU platform in double precision.

헬리콥터에서 발사되는 발사체의 궤적 예측

A program that predicts trajectories of projectiles influenced by the interference flow field of helicopters is developed. The interference flow field are computed using a compressible inviscid solver in conjunction with an actuator disc model. The trajectories are predicted using 6-DOF (Degree of Freedom) equations as well as an alternative form of modified point mass equations of motion. The method for the interference flow field prediction method are validated with ROBIN(ROtor Body INteraction) model. A Sierra international bullet and a 105mm projectile are used to validate the trajectory method. Trajectories of a Sierra International bullet and a HYDRA 70 rocket firing from a helicopter are predicted.

An immersed boundary method for simulation of inviscid compressible flows

SUMMARYIn this paper, an immersed boundary method for simulating inviscid compressible flows governed by Euler equations is presented. All the mesh points are classified as interior computed points, immersed boundary points (interior points closest to the solid boundary), and exterior points that are blanked out of computation. The flow variables at an immersed boundary point are determined via the approximate form of solution in the direction normal to the wall boundary. The normal velocity is evaluated by applying the no‐penetration boundary condition, and therefore, the influence of solid wall in the inviscid flow is taken into account. The pressure is computed with the local simplified momentum equation, and the density and the tangential velocity are evaluated by using the constant‐entropy relation and the constant‐total‐enthalpy relation, respectively. With a local coordinate system, the present method has been extended easily to the three‐dimensional case. The present work is the first endeavor to extend the idea of hybrid Cartesian/immersed boundary approach to compressible inviscid flows. The tedious task of handling multi‐valued points can be eliminated, and the overshoot resulting from the extrapolation for the evaluation of flow variables at exterior points can also be avoided. In order to validate the present method, inviscid compressible flows over fixed and moving bodies have been simulated. All the obtained numerical results show good agreement with available data in the literature. Copyright © 2014 John Wiley & Sons, Ltd.

Higher-order multi-dimensional limiting strategy for discontinuous Galerkin methods in compressible inviscid and viscous flows

This paper deals with the multi-dimensional limiting process (MLP) for discontinuous Galerkin (DG) methods to compute compressible inviscid and viscous flows. The MLP, which has been quite successful in finite volume methods (FVM), is extended to DG methods for hyperbolic conservation laws. In previous works, the MLP was shown to possess several superior characteristics, such as the ability to control multi-dimensional oscillation efficiently and to capture both discontinuous and continuous multi-dimensional flow features accurately within the finite volume framework. In particular, the oscillation-control mechanism in multiple dimensions was established by combining the local maximum principle and the multi-dimensional limiting (MLP) condition, leading to the formulation of efficient and accurate MLP-u slope limiters.The MLP limiting strategy is now extended to the higher-order DG framework to develop the hierarchical MLP formulation on unstructured grids, which facilitates the capturing of compressible flow structures very accurately. By combining the behavior of local extrema with the augmented MLP condition, the hierarchical MLP method (the MLP-based troubled-cell markers and the MLP limiters in the RKDG formulation) is developed for arbitrary DG-Pn polynomial approximations. Through extensive numerical analyses and computations on unstructured grids, it is demonstrated that the proposed hierarchical DG-MLP method yields outstanding performance in resolving non-compressive as well as compressive flow features.

Circular function-based gas-kinetic scheme for simulation of inviscid compressible flows

This paper presents a new gas-kinetic scheme for simulation of compressible inviscid flows. It starts to simplify the integral domain of Maxwellian distribution function over the phase velocity ξ and phase energy ζ to the integral domain of modified Maxwellian function over the phase velocity ξ only. The influence of integral over phase energy ζ is embodied as the particle internal energy ep. The modified Maxwellian function is further simplified to a circular function with the assumption that all the particles are concentrated on a circle. Then two circular function-based gas-kinetic schemes are presented for simulation of compressible inviscid flows. In the new schemes, no error and exponential functions, which are often appeared in the Maxwellian function-based gas-kinetic schemes, are involved. As a result, the new schemes can be implemented in a more efficient way. To validate the proposed new gas-kinetic schemes, test examples in the transonic flow, supersonic flow and hypersonic flow regimes are solved. Numerical results showed that the solution accuracy of the circular function-based gas-kinetic schemes is comparable to that of corresponding Maxwellian function-based gas-kinetic schemes. However, the circular function-based gas-kinetic schemes need less computational effort.

Numerical Solution of Multidimensional Compressible Flow by High Order Nodal Discontinuous Galerkin Method

In this paper we present a robust, high order method for numerical solution of multidimensional compressible inviscid flow equations. Our scheme is based on Nodal Discontinuous Galerkin Finite Element Method (NDG-FEM). This method utilizes the favorable features of Finite Volume Method (FVM) and Finite Element Method (FEM). In this method, space discretization is carried out by finite element discontinuous approximations. The resulting semi discrete differential equations were solved using explicit Runge-Kutta (ERK) method. In order to compute fluxes at element interfaces, we have used Roe Approximate scheme. In this article, we demonstrate the use of exponential filter to remove Gibbs oscillations near the shock waves. Numerical predictions for two dimensional compressible fluid flows are presented here. The solution was obtained with overall order of accuracy of 3. The numerical results obtained are compared with experimental and finite volume method results.