An efficient algorithm for computing smoothness indicators for WENO schemes

WENO schemes are a popular class of shock-capturing schemes which adopt an adaptive-stencil approach to interpolation. WENO schemes rely on smoothness indicators to assess the relative smoothness of the solution within the sub-stencils. Computing these smoothness indicators is the most expensive operation in the WENO reconstruction procedure. In this paper, an efficient algorithm is proposed to compute these quantities without sacrificing the positivity property of the smoothness indicators. The proposed algorithm involves linear combinations of the undivided differences which can be computed efficiently in a recursive manner. This allows the computation of the smoothness indicators to be performed using significantly fewer floating-point operations compared to conventional implementations. Moreover, the proposed algorithm is simple to implement and involves fewer constants

Some Insights into the Screech Tone of Under-Expanded Supersonic Jets Using Dynamic Mode Decomposition

Jet screech is an intense pure tone which has attracted decades of research interest due to its possible detrimental effect on engineering structures. Its modes and closure mechanisms have been investigated analytically, experimentally, and numerically; however, there are still outstanding questions regarding the generation and propagation of instabilities in the near-field region. Recent studies have identified that these instabilities travel inside the jet potential during the screech process to form the complete feedback loop. Using dynamic mode decomposition on a three-dimensional pressure near field from large-eddy simulation results, the present study examines the viability of modal decomposition to provide further insights into the screech mode and its associated characteristics, and investigates the effect of temperature mixing in jet screech. The results show that modal decomposition identifies the helical structure of screech mode. Furthermore, a method is proposed to reveal the temporal evolution of dynamic screech mode. It was found that the bulk behavior of the pressure field at screech frequency propagates backward toward the nozzle exit.Ministry of Education (MOE)The authors gratefully acknowledge the support provided for this study by the Singapore Ministry of Education AcRF Tier-2 Grant (Grant No. MOE2014-T2-1-002)

A carbuncle cure for the Harten-Lax-van Leer contact (HLLC) scheme using a novel velocity-based sensor

AbstractA hybrid numerical flux scheme is proposed by adapting the carbuncle-free modified Harten-Lax-van Leer contact (HLLCM) scheme to smoothly revert to the Harten-Lax-van Leer contact (HLLC) scheme in regions of shear. This hybrid scheme, referred to as the HLLCT scheme, employs a novel, velocity-based shear sensor. In contrast to the non-local pressure-based shock sensors often used in carbuncle cures, the proposed shear sensor can be computed in a localized manner meaning that the HLLCT scheme can be easily introduced into existing codes without having to implement additional data structures. Through numerical experiments, it is shown that the HLLCT scheme is able to resolve shear layers accurately without succumbing to the shock instability.</jats:p

Exact solutions to the Navier - Stokes equation from the interpretation of the Schrodinger wave function

Navier-Stokes equation is considered to be the fundamental equation governing the dynamics of fluids. However, exact analytical solutions to this equation could only be obtained for highly idealized flows. More general exact solutions, if found, would be greatly beneficial since they would further our understanding of fluids and could be used to improve current computational solvers. In the recent years, a general solution to the Navier-Stokes equation for incompressible fluids has been presented (V. Kulish & Lage, 2002; V. V. Kulish & Lage, 2013) based on a quantum mechanics formulation known as quantum fluid dynamics (QFD). QFD predicts the presence of a velocity potential in the motion of quantum particles. This result was used to argue the presence of a velocity potential in fluid motion. The objective of the current study is to verify the validity of this general solution mathematically and conceptually. Through direct substitution, the mathematical validity of the general solution was confirmed provided a velocity potential function exists for the fluid motion. However, the general solution failed to produce the right solutions even for simple flows. The reason for this apparent failure requires further analysis. Conceptually, the existence of a velocity potential function for fluid motion cannot be argued from QFD as was done in the papers. Although not applicable to conventional fluids, the solution may find applications to the seemingly inviscid superfluids which are believed to possess a velocity potential function.Bachelor of Engineering (Aerospace Engineering

Development of high order WENO schemes for large-eddy simulation of compressible flows in OpenFOAM

Performing LES of compressible flows is a challenging affair. On the one hand, one strives to minimize numerical dissipation to preserve ‘all’ scales of motions. On the other hand, introducing some dissipation is necessary to guarantee numerical stability near sharp gradients and to capture shocks. High order shock-capturing schemes are able to meet these contradicting demands and WENO schemes are one class of such schemes that has found great success in the past few decades. Unfortunately, most, if not all, commercial software still rely on second order methods and do not allow the incorporation of new discretization schemes. While individual research groups have developed sophisticated high order schemes, these are proprietary in-house codes. Developing an in-house code from scratch is a daunting process but the proliferation of opensource code has opened up another avenue for the development of new CFD codes. In the recent years, OpenFOAM, an opensource CFD toolkit, has been extensively used in the development of customized numerical schemes and solvers. However, the most popular of its compressible flow solvers, rhoCentralFoam, is hardly suitable for LES since it uses TVD schemes for shock-capturing which revert to being only first order accurate in non-monotone regions. Therefore, the overall aim of the present work was to implement high order WENO schemes in OpenFOAM and construct a transient compressible flow solver for performing LES of compressible flow. With this goal in mind, an arbitrary order WENO scheme capable of handling non-uniform meshes and moderate non-orthogonality has been implemented in OpenFOAM as a standalone library. The scheme is implemented in a localized, face-based manner (dimensional split approach) for efficiency and robustness. The computation of sub-stencil smoothness indictors is implemented in a general, positive semi-definite form using polynomial coefficients rather being hardcoded using explicit expressions. A transient flow solver has also been developed to run in conjunction with the WENO library. The solver employs a novel hybrid flux scheme which exploits the efficient computation of smoothness indicators to achieve significant speedups. The accuracy and efficiency of the WENO schemes and the hybrid solver have been demonstrated using suitable linear advection and Euler test cases. To reduce the numerical dissipation of the WENO schemes further, a new adaptive mapping procedure, sensitive to the local smoothness of the solution, was designed based on a new family of mapping functions g_{\text{IM+}}\left(\omega; k, m, s\right). A parametric study was performed to determine suitable values of the parameters. The resultant adaptive mapped WENO scheme, referred to as WENO-AIM(4,2,1e4), has been demonstrated to outperform many existing improved WENO schemes in several benchmark test cases. Finally, the hybrid solver was used to perform large-scale LES of under-expanded supersonic jets using seventh order WENO-AIM(4,2,1e4) scheme. The results have been found to be in good agreement with experimental schlieren images and microphone measurements. In particular, the frequency and amplitude of screech tone at NPR=5 was found to match experimental results well.Doctor of Philosoph

A large-eddy simulation study on vortex-ring collisions upon round cylinders

A large-eddy simulation based numerical study was conducted on head-on collisions between vortex-rings and round cylinders. The vortex-ring Reynolds number was Re = 4000, while the ratio of the cylinder diameter to vortex-ring diameter (i.e., diameter ratio, D/d) was varied from 4 to 1. Vortical behavior predicted by the present simulations is observed to agree well with an earlier experimental study [New, T. H., and Zang, B., “Head-on collisions of vortex rings upon round cylinders,” J. Fluid Mech. 833, 648 (2017)]. The present simulations also reveal additional flow details on the vortex dynamics and vortex-core trajectories, which have not been observed previously. First, vortex-dipoles produced by D/d ≤ 2 cylinders are cross sections of elliptic vortex-ringlets formed via vortex disconnection/reconnection of secondary vortex-ring segments. Second, the aspect ratio of the elliptic vortex-ringlets increases when a smaller diameter-ratio cylinder is used, and finally, they undergo axis-switching behavior. Furthermore, up to three sets of tertiary vortex-ring cores are formed along the D/d = 2 and 1 cylinder straight-edges where they subsequently merge with the secondary vortex-ring cores within the confines of the primary vortex-ring cores. This merged vortex core moves toward the collision axis and forms an inner vortex-dipole with a wall separated vortex. Along the convex surface, up to two sets of tertiary vortex-ring cores are observed for D/d = 2 and 1 cylinders, and trajectories of the vortex-dipoles agree well with the past experimental results. These observations support the notion that higher vortex-stretching levels resulting from the use of small diameter-ratio cylinders with higher surface curvatures underpin the wide range of vortical behavior observed here.Submitted/Accepted versio

A domain decomposition technique for small amplitude wave interactions with shock waves

In this paper, a domain decomposition technique in the finite volume framework is presented to propagate small amplitude acoustic and entropy waves in a linearized Euler region and simulate the interaction of these waves with an initially steady normal shock in a nonlinear region. An overset method is used to two-way couple the linear and nonlinear regions that overlap each other. Linearized solvers alone cannot capture this interaction due to the discontinuity encountered at shocks. On the other hand, nonlinear solvers based on second order shock-capturing schemes will result in excessive dissipation and dispersion for the small disturbances. The domain decomposition technique provides a good balance between minimizing dissipation and dispersion errors while enabling nonlinear shock-acoustic interactions. To preserve low dispersion and dissipation, a DRP scheme is used to simulate the incoming and outgoing waves in the linear region. To capture the shock wave interaction and motion, a hybrid central-upwind flux scheme is used in the nonlinear region that contains the shock. Grid sensitivity studies for an acoustic wave propagating in stationary flow were performed to compare the linear, nonlinear, and domain decomposition solvers. The nonlinear solver required ten times the mesh resolution to achieve similar accuracy as the linear solver, resulting in a forty-fold increase in computational time. For modest cell size ratios, the domain decomposition solver reduced the computational time by a factor of three compared to the nonlinear solver while achieving similar accuracy. Interaction of standing shocks with acoustic and entropy waves of amplitudes ϵ=±10−2 and ±10−5 was investigated using the domain decomposition technique. The numerical results for ϵ=±10−2 compared well with the linearized interaction analysis (LIA) with less than 3% discrepancy in terms of the amplification factors. The domain decomposition technique acts as a low pass filter that averages the post-shock oscillations generated by the slow-moving shocks in the nonlinear region, resulting in the correct amplification factors in the linear region. For the smaller amplitudes of ϵ=±10−5, the amplification factors deviated from LIA predictions by up to 70%. Numerical results suggest that the large discrepancy for the small amplitude cases is due to insufficient mesh resolution for capturing extremely slow-moving shocks.Ministry of Education (MOE)Submitted/Accepted versionThis research is supported by Ministry of Education, Singapore, under its Academic Research Fund Tier 1 (RG183/18)

Adaptive mapping for high order WENO methods

In this paper, a novel mapping approach through the use of adaptive mapping functions is introduced for high order weighted essentially non-oscillatory (WENO) methods. The new class of adaptive mapping functions are designed to adjust themselves to the solution based on a simple parameter calculated using the smoothness indicators that are readily available during computation. It is shown that this adaptive nature allows the resultant mapped WENO scheme to maintain sub-stencil weights close to the optimal weights in smooth regions without amplifying the weights of non-smooth stencils containing discontinuities. Therefore, adaptive mapping achieves enhanced accuracy in smooth regions and is more resistant against spurious oscillations near discontinuities. Taylor series analysis of the seventh order finite volume WENO scheme has been performed to demonstrate the loss of accuracy of the original WENO method near critical points. The convergence rates of the seventh order finite volume WENO scheme with adaptive mapping have been shown through a simple numerical example. Excellent results have been obtained for one-dimensional linear advection cases especially over long output times. Improved results have also been obtained for one- and two-dimensional Euler equation test cases.Ministry of Education (MOE)Nanyang Technological UniversityNational Supercomputing Centre (NSCC) SingaporeAccepted versionThe authors gratefully acknowledge the support for the present work by Singapore Ministry of Education AcRF Tier-2 grant (MOE2014-T2-1-002), National Supercomputing Center Singapore and support for the first author through Graduate Research Officer scholarship from the School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore

A New Mapped WENO Method for Hyperbolic Problems

In this study, a new family of rational mapping functions gRM(ω;k,m,s) is introduced for seventh-order WENO schemes. gRM is a more general family of mapping functions, which includes other mapping functions such as gM and gIM as special cases. The mapped WENO scheme WENO-IM(2,0.1), which uses gIM, performs excellently at fifth order but rather poorly at seventh order. The reason for this loss of accuracy was found to be the over-amplification of very small weights by the mapping process, which can be traced back to the large slope of gIM at ω = 0. For m > 1, gRM can be designed to have a unit slope at ω = 0, which will preserve small weights with little to no amplification. It has been demonstrated through several one-dimensional linear advection test cases that the mapped WENO scheme WENO-RM(6,3,2 × 103), which uses the mapping function gRM(ω;6,3,2 × 103), outperforms both WENO-M and WENO-IM(2,0.1) at seventh order. The proposed scheme also performs better at a number of one-dimensional inviscid gas flow problems compared to other popular WENO schemes such as the WENO-Z scheme