Two-dimensional Compressible Navier-Stokes Equations Research Articles

Modification in airfoil’s tonal noise using periodic suction-blowing excitation

This paper examines the modifications in the flow and sound fields due to the application of a periodic suction-blowing excitation strip on the top and bottom surfaces of a NACA0012 airfoil. The flow and sound fields have been directly computed (DNS approach) by solving unsteady, two-dimensional compressible Navier–Stokes equations using physical dispersion relation preserving schemes. The flow approaches the airfoil at a Reynolds number of 5000 and 5o angle of attack. In the no-excitation case, the sound is mainly created by the lift and drag dipole equivalent sources associated with vortex shedding. However, the application of suction-blowing excitation results in two additional sound sources: monopole-equivalent sound sources due to fluid movement in and out of the airfoil surface and the drag dipole-equivalent sound sources due to periodic change in the airfoil’s effective cross-sectional area. The combined effects of these sound sources result in significant mitigation of sound on the suction side of the airfoil. The excitation amplitude and forcing frequency effects on sound field modifications have been studied in depth. Beats of sound have been observed when the forcing frequency is close to the excitation frequency.

Direct numerical simulations of long-range infrasound propagation: Implications for source spectra estimation.

The evolution of observed dominant frequencies from a high-intensity infrasonic pulse with receiver range and stratospheric temperature is investigated using direct numerical simulations of the two-dimensional unsteady compressible Navier-Stokes equations. There is a high level of uncertainty in estimating source dominant frequencies based on received signals at sparse points on the ground. Nonlinear propagation effects in the ground-level thermospheric arrivals are found to significantly alter dominant frequency measurements compared to stratospheric arrivals with smaller amplitude sources. With a larger amplitude source, variations in observations are minimized as a result of nonlinear effects being ubiquitous across all atmospheric components of received signals but have a greater offset to the source dominant frequency. An approach to determine the source dominant frequency and minimize atmospheric variability is presented by calculating a source-to-receiver spectral transfer function averaged across the atmospheric states. This method reduces atmospheric variability in source frequency estimates within the pseudo-linear propagation regime and the average error to the known source frequency with a large amplitude source. The reduction of errors in source frequency estimates demonstrates the feasibility of using remote infrasound measurements as an indicator of source frequency and, in turn, the explosive yield of clandestine nuclear weapon test explosions.

Global solutions of 2D isentropic compressible Navier-Stokes equations with one slow variable

Motivated by [2,3], we prove the global existence of solutions to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data which are slowly varying in one direction and with initial density being away from vacuum. In particular, we present examples of initial data which generate unique global smooth solutions to 2D compressible Navier-Stokes equations with constant viscosity and with initial data which are neither small perturbation of some constant equilibrium state nor of small energy.

角柱近傍を移動する渦対による音波発生について

The sound generated by a vortex pair moving close to a rectangular cylinder is studied by high-resolution simulations of the two-dimensional unsteady compressible Navier-Stokes equations at Mach numbers ranging from 0.05 to 0.3. In this study, the Mach number Ma is defined by Ma=Vt /c0, where Vt is the translation velocity of the co-moving vortex pair and c0 denotes the speed of sound. The directly computed far-field sound is compared to the prediction of the acoustic analogy proposed by Lighthill (1952) and Curle (1955). The prediction is in excellent agreement with the simulation, where the two-dimensional form of Curle’s modified solution is adopted for an accurate calculation of the sound waves with a phase difference between various vortices around the cylinder. It could be confirmed that the scaling law in two dimensions holds in this simulation. In this study, the sound generation due to the dipole and quadrupole fields is investigated by calculating the two terms in Curle’s analogy. Examining the surface distribution of dipoles on the cylinder indicates that the force exerted on the fluid by the solid boundaries of the side/upper wall of the rectangular cylinder generates the sound when the co-translating vortices approach /pass the rectangular cylinder, which is the sound due to dipoles corresponding to the directivity of the sound in the polar diagrams. The area distribution of quadrupoles which is the second time derivative of the area integral of Lighthill’s stress tensor indicates that the sound is emitted from a vortex generated at the corner of the rectangular cylinder when the vortex pair approaches the cylinder and where the interaction between many vortices around the cylinder and the co-translating vortices generates the sound when the co-translating vortices pass the cylinder.

Adaptive sparse interpolation for accelerating nonlinear stochastic reduced-order modeling with time-dependent bases

Stochastic reduced-order modeling based on time-dependent bases (TDBs) has proven successful for extracting and exploiting low-dimensional manifold from stochastic partial differential equations (SPDEs). The nominal computational cost of solving a rank-r reduced-order model (ROM) based on time-dependent basis, a.k.a. TDB-ROM, is roughly equal to that of solving the full-order model for r random samples. As of now, this nominal performance can only be achieved for linear or quadratic SPDEs—at the expense of a highly intrusive process. On the other hand, for problems with non-polynomial nonlinearity, the computational cost of solving the TDB evolution equations is the same as solving the full-order model. In this work, we present an adaptive sparse interpolation algorithm that enables stochastic TDB-ROMs to achieve nominal computational cost for generic nonlinear SPDEs. Our algorithm constructs a low-rank approximation for the right hand side of the SPDE using the discrete empirical interpolation method (DEIM). The presented algorithm does not require any offline computation and as a result the low-rank approximation can adapt to any transient changes of the dynamics on the fly. We also propose a rank-adaptive strategy to control the error of the sparse interpolation. Our algorithm achieves computational speedup by adaptive sampling of the state and random spaces. We illustrate the efficiency of our approach for two test cases: (1) one-dimensional stochastic Burgers’ equation, and (2) two-dimensional compressible Navier–Stokes equations subject to one-hundred-dimensional random perturbations. In all cases, the presented algorithm results in orders of magnitude reduction in the computational cost.

A new direct discontinuous Galerkin method with interface correction for two-dimensional compressible Navier-Stokes equations

We propose a new formula for the nonlinear viscous numerical flux and extend the direct discontinuous Galerkin method with interface correction (DDGIC) of Liu and Yan [1] to compressible Navier-Stokes equations. The new DDGIC framework is based on the observation that the nonlinear diffusion can be represented as a sum of multiple individual diffusion processes corresponding to each conserved variable. A set of direction vectors corresponding to each individual diffusion process is defined as a function of the exact solution and approximated using the average value of the numerical solution at the cell interfaces. The new framework only requires the computation of conserved variables' gradient, which is linear and approximated by the original direct DG numerical flux formula. The proposed method greatly simplifies the implementation, and thus, can be easily extended to general equations and turbulence models. Numerical experiments with P1, P2, P3 and P4 polynomial approximations are performed to verify the optimal (k+1)th high-order accuracy of the method. The new DDGIC method is shown to be able to accurately calculate physical quantities such as lift, drag, and friction coefficients as well as separation angle and Strouhal number.

Ill-posedness for the Cauchy problem of the two-dimensional compressible Navier-Stokes equations for an ideal gas

Ill-posedness for the Cauchy problem of the two-dimensional compressible Navier-Stokes equations for an ideal gas

Feature extraction of fields of fluid dynamics data using sparse convolutional autoencoder

A neural network technique that extracts underlying flow features from the original flow field data is newly proposed. The technique here is based on the convolutional and sparse autoencoder learning algorithms and is called sparse convolutional autoencoder. Unlike the typical convolutional neural network (CNN) that changes the size of the data itself in the intermediate layers, flow field data size is not changed in the learning process of this method and only the numbers of channels are changed in each layer. Different but the same size of the data as the input are obtained by convolution with multiple spatially overlapping flow field data under the assumption of sparsity. When data restoration is realized in this autoencoder system, the channel numbers of data in the intermediate layers turn out to contain different flow characteristics of the original flow field. The proposed method is applied to the low Reynolds number flows over a circular cylinder. The high-fidelity unsteady flow data obtained by solving two-dimensional compressible Navier–Stokes equations with a high-resolution numerical scheme are used as a test case. In the proposed method, sparsity introduced in the middle-hidden layer is essential for the successful separation of the original data. The results presented in the example seem to correspond to positive and negative magnitudes of the original data, but future studies will reveal other features of the method. The present method shows flow features different from those of proper orthogonal decomposition in each mode, which is probably due to nonlinear decomposition in the CNN process.

Windowed space–time least-squares Petrov–Galerkin model order reduction for nonlinear dynamical systems

Projection-based reduced-order models (PROMs) restrict the full-order model (FOM) to a low-dimensional subspace. Space–time PROMs in particular restrict the FOM to a temporally space–time trial subspace, and compute approximate solutions through a residual orthogonalization or minimization process. One such technique of interest is the space–time least-squares Petrov–Galerkin method (ST-LSPG), which reduces both the spatial and temporal dimensions. However, ST-LSPG is computationally expensive, because it requires solving a dense space–time system with a space–time basis that is calculated over the entire global time domain, which can be unfeasible for large-scale applications. To address these challenges, this paper presents the windowed space–time least-squares Petrov–Galerkin method (WST-LSPG) for model reduction of nonlinear parameterized dynamical systems. The proposed WST-LSPG approach addresses the aforementioned challenges by (1) dividing the time simulation into windows, (2) devising a unique low-dimensional high-fidelity space–time trial subspace for each window, and (3) minimizing the discrete-in-time space–time residual of the dynamical system over each window. In this formulation, the problem confines coupling within each window, but solves space–time residual minimization problems sequentially across the windows. WST-LSPG is equipped with hyper-reduction techniques to further reduce the computational cost. Numerical experiments for the one-dimensional Burgers’ equation and the two-dimensional compressible Navier–Stokes equations for flow over a NACA 0012 airfoil demonstrate that WST-LSPG is superior to ST-LSPG in terms of accuracy and computational gain by as much as 99%.

Initial boundary value problems for two-dimensional isentropic compressible Navier-Stokes equations with rotating effect terms

We consider the initial boundary value problem for two-dimensional isentropic compressible Navier-Stokes equations with rotating effect terms. This problem is related to the motion of the compressible viscous flow surrounding a rotating obstacle. We show the global existence and uniqueness of strong solutions with initial data close to some approximate solution, provided that the friction coefficient α attached to the Navier-slip boundary condition is small. The large time behavior of the solutions is also studied.

Global large solutions to the two-dimensional compressible Navier–Stokes equations

We obtain the global large solutions to the compressible Navier–Stokes equations in \({\mathbb {R}}^2\). The solution is large in the sense that there is no smallness assumption applied to one component of the initial incompressible velocity.

Spatio-temporal aeroacoustic–structural responses of cavity-backed elastic panel liner exposed to grazing duct flow

An in-duct device laid with acoustic liner is a popular noise control option for extensive ductworks carrying unsteady flow in various engineering applications. Besides the conventional absorptive liner, elastic panel liner backed by a cavity emerges as a promising alternative concept that has advantages in providing effective broadband reduction at low frequencies yet with significantly low fluid dynamic pressure loss. The present paper reports a numerical study of aeroacoustic–structural interaction of elastic panel liner exposed to subsonic duct boundary layer flow in time domain and explores its role in liner mitigation of plane wave broadband acoustic excitation. The aeroacoustics of the duct flow and panel dynamics are respectively modeled by the two-dimensional compressible Navier–Stokes equations that solved by the conservation element and solution element (CE/SE) method, and one-dimensional elastic panel equation that is solved with finite difference method. The interaction between aeroacoustics and panel dynamics is resolved with a monolithic coupling scheme that is well validated down to acoustic scales. Special attention is placed on the sensitivity of liner acoustic performance to duct flow Mach number (M≤0.3) and the relative directions between duct flow and acoustic excitation. Extensive cross-spectral analyses of acoustic results reveal that the transmission loss of the elastic panel liner is primarily a result of the destructive interference of the incident wave with the scattered wave as well as the dissipation of non-planar waves radiated by the panel vibration. The contribution of interference and dissipation to liner reflection and absorption are particularly pronounced at low and high frequencies respectively. In addition, the effect of flow is not strong for M≤0.1. The presence of flow with the incident wave in the flow direction mainly weakens the reflection. However, the effect of flow with incident waves against the flow largely enhances the absorption. Overall, the elastic panel liner is much more effective when the noise source radiates against the flow.

Immersed boundary conditions for hypersonic flows using ENO-like least-square reconstruction

In this study, we present an immersed boundary reconstruction method for the simulation of viscous compressible flows presenting strong discontinuities in the vicinity of the immersed body. The technique we propose is based on a weighted least-square reconstruction. Its major feature is that the weights are automatically adapted to the presence of a discontinuity within the least-square stencil by using a shock-sensor approach. In so doing, the reconstruction becomes ENO-like and mathematically consistent even in the presence of strong discontinuities. The objective of this paper is to demonstrate that it then becomes possible to use an immersed boundary method for the simulation of hypersonic flows and that it is robust, even for flows around complex geometries. To do so, two-dimensional compressible Navier-Stokes equations are solved to study canonical hypersonic flows and it is showed that the results are in good agreement with either experimental data and/or body-fitted approaches and that the impact of the immersed boundary technique on the overall spatial order of convergence is reasonable.

Evaluation of breakup models for liquid side jets in supersonic cross flows

In this study, numerical simulations of the interaction between supersonic crossflow and a side liquid jet were performed. The processes of the liquid jet breakup, flow separation and shock waves resulting from the interaction of the gas flows and liquid droplets were predicted. In the numerical simulation, a two-way coupling Eulerian-Lagrangian method was used. The two-dimensional unsteady compressible Navier-Stokes Equations were solved based on the MUSCL type (Monotone Upstream-Centred Scheme for Conservation Laws) spatial differencing coupling with Roe-type flux-difference splitting and a third-order accurate Runge-Kutta method for time-splitting. The atomization of liquid droplets was simulated using the combined KH-RT breakup model. The modified breakup model considered both the interface instability and the sudden deformation of a liquid spray induced by the compressible flows. Shock waves, recirculation zones and breakup processes of droplet particles were captured. The predicted droplet distributions at various operating conditions were discussed. Satisfied agreement between simulations and measurements is shown in the current work.

A Continuation Principle to the Cauchy Problem of Two-Dimensional Compressible Navier-Stokes Equations with Variable Viscosity

This paper aims to investigate effectivity problems of pluricanonical systems on varieties of general type in positive characteristic. In practice, we will consider a sub-linear system $|S^0_{-}(X, K_X + nK_X)| \subseteq |H^0(X, K_X +nK_X)|$ generated by certain Frobenius stable sections, and prove that for a minimal terminal threefold $X$ of general type with either $q(X)>0$ or Gorenstein singularities, if $n\geq 28$ then $|S^0_{-}(X, K_X + nK_X)| \neq \emptyset$; if $n\geq 42$ then the linear system $|S^0_{-}(X, K_X + nK_X)|$ defines a birational map.

Vanishing viscosity limit to the planar rarefaction wave for the two-dimensional full compressible Navier-Stokes equations

The vanishing viscosity limit for the multi-dimensional compressible Navier-Stokes equations to the corresponding Euler equations is a difficult and challenging problem in the mathematics. Recently, L.Li, D.Wang and Y.Wang [17] verified that the solutions for the 2D compressible isentropic Navier-Stokes equations converge to the planar rarefaction wave solution for the corresponding 2D Euler equations as viscosity vanishes with a convergence rate ϵ1/6|ln⁡ϵ|. In this paper, the vanishing viscosity limit of 2D non-isentropic compressible Navier-Stokes equations is studied. In contrast to the work [17], the convergence rate for the 2D full compressible Navier-Stokes equations is improved to ϵ2/7|ln⁡ϵ|2 by choosing a different scaling argument and performing more detailed energy estimates.

$hp$-Multilevel Monte Carlo Methods for Uncertainty Quantification of Compressible Navier--Stokes Equations

We propose a novel $hp$-multilevel Monte Carlo method for the quantification of uncertainties in the compressible Navier-Stokes equations, using the Discontinuous Galerkin method as deterministic solver. The multilevel approach exploits hierarchies of uniformly refined meshes while simultaneously increasing the polynomial degree of the ansatz space. It allows for a very large range of resolutions in the physical space and thus an efficient decrease of the statistical error. We prove that the overall complexity of the $hp$-multilevel Monte Carlo method to compute the mean field with prescribed accuracy is, in best-case, of quadratic order with respect to the accuracy. We also propose a novel and simple approach to estimate a lower confidence bound for the optimal number of samples per level, which helps to prevent overestimating these quantities. The method is in particular designed for application on queue-based computing systems, where it is desirable to compute a large number of samples during one iteration, without overestimating the optimal number of samples. Our theoretical results are verified by numerical experiments for the two-dimensional compressible Navier-Stokes equations. In particular we consider a cavity flow problem from computational acoustics, demonstrating that the method is suitable to handle complex engineering problems.

Vanishing Viscosity Limit to the Planar Rarefaction Wave for the Two-Dimensional Compressible Navier–Stokes Equations

The vanishing viscosity limit of the two-dimensional (2D) compressible isentropic Navier-Stokes equations is studied in the case that the corresponding 2D inviscid Euler equations admit a planar rarefaction wave solution. It is proved that there exists a family of smooth solutions for the 2D compressible Navier-Stokes equations converging to the planar rarefaction wave solution with arbitrary strength for the 2D Euler equations. A uniform convergence rate is obtained in terms of the viscosity coefficients away from the initial time. In the proof, the hyperbolic wave is crucially introduced to recover the physical viscosities of the inviscid rarefaction wave profile, in order to rigorously justify the vanishing viscosity limit.

Global strong solutions to the two-dimensional full compressible Navier-Stokes equations with vacuum

In this paper, we consider the global existence of strong solutions to the two-dimensional viscous, compressible, and heat conducting Navier-Stokes equations on the square domain Ω=[0,1]2. Based on the blow-up criterion and uniform estimates, we prove that the strong solution exists globally in time if the initial mass is small for the constant coefficients of viscosity and heat conductivity.

Study of Interaction between Unsteady Supersonic Jet and Vortex Rings Discharged from Elliptical Cell

The unsteady supersonic jet formed by a shock tube with a small high-pressure chamber was used as a simple alternative model for pulsed laser ablation. Understanding the vortex ring formed by the shock wave is crucial in clarifying behavior of unsteady supersonic jet discharged from elliptical cell. The purpose of this study has been to reveal behavior of the supersonic jet and the vortex rings. The unsteady behavior of a flow is investigated numerically by solving the axisymmetric two-dimensional compressible Navier-Stokes equations. The system of the calculation is a model of laser ablation of a certain duration followed by discharge through the cell exit. The parameters for the calculations are the pressure ratio of the shock tube and the cell diameter. As a result, it was found that the diameter of the elliptical cell had the effect on the interaction between vortex rings and the jet.