
[Sponsors] 
Publication date: 30 October 2023
Source: Computers & Fluids, Volume 265
Author(s): Chao Ma, Jie Wu, Xiangyu Gu, Liming Yang
Publication date: 30 October 2023
Source: Computers & Fluids, Volume 265
Author(s): T. Fabbri, G. Balarac, V. Moureau, P. Benard
Publication date: 30 October 2023
Source: Computers & Fluids, Volume 265
Author(s): Lin Liu, Siyu Chen, Libo Feng, Jing Zhu, Jiangshan Zhang, Liancun Zheng, Chiyu Xie
Publication date: 30 October 2023
Source: Computers & Fluids, Volume 265
Author(s): Ral Bielawski, Shivam Barwey, Supraj Prakash, Venkat Raman
Publication date: 30 October 2023
Source: Computers & Fluids, Volume 265
Author(s): Jiabao Chen, Yan Wang, Dangguo Yang, Qing Chen, Jianhong Sun
Publication date: 30 October 2023
Source: Computers & Fluids, Volume 265
Author(s): JiWang Luo, Li Chen, Yang Xia, Xinjian Zheng, WenQuan Tao
Publication date: 30 October 2023
Source: Computers & Fluids, Volume 265
Author(s): Jianan Zeng, Ruifeng Yuan, Yanbing Zhang, Qi Li, Lei Wu
Publication date: 30 October 2023
Source: Computers & Fluids, Volume 265
Author(s): Fuyu Zhao, Cheng Wang, Xiyu Jia, Wanli Wang
Publication date: 30 October 2023
Source: Computers & Fluids, Volume 265
Author(s): Hannes Mandler, Bernhard Weigand
Publication date: 30 October 2023
Source: Computers & Fluids, Volume 265
Author(s): Saumitra Joshi, Jiaqing Kou, Aurelio Hurtado de Mendoza, Kunal Puri, Charles Hirsch, Gonzalo Rubio, Esteban Ferrer
Obtaining accurate reducedorder models (ROMs) of convectiondominated problems can be very challenging because they generally have a slowly decaying Kolmogorov n$$ n $$width. In this work, we develop a costefficient method that combines both highdimensional model and ROM evaluations by exploiting the local spatial coherence of these problems. We obtain up to a factor of five speedup for predictive, shock dominated flow simulations.
The vast majority of reducedorder models (ROMs) first obtain a low dimensional representation of the problem from highdimensional model (HDM) training data which is afterwards used to obtain a system of reduced complexity. Unfortunately, convectiondominated problems generally have a slowly decaying Kolmogorov n$$ n $$width, which makes obtaining an accurate ROM built solely from training data very challenging. The accuracy of a ROM can be improved through enrichment with HDM solutions; however, due to the large computational expense of HDM evaluations for complex problems, they can only be used parsimoniously to obtain relevant computational savings. In this work, we exploit the local spatial coherence often exhibited by these problems to derive an accurate, costefficient approach that repeatedly combines HDM and ROM evaluations without a separate training phase. Our approach obtains solutions at a given time step by either fully solving the HDM or by combining partial HDM and ROM solves. A dynamic sampling procedure identifies regions that require the HDM solution for global accuracy and the reminder of the flow is reconstructed using the ROM. Moreover, solutions combining both HDM and ROM solves use spatial filtering to eliminate potential spurious oscillations that may develop. We test the proposed method on inviscid compressible flow problems and demonstrate speedups up to a factor of five.
Extended modal discontinuous Galerkin scheme using Legendre and Laguerre basis functions provides accurate simulation of wave motions and diffusive processes in multidimensional unbounded domains. For the same accuracy level, the extended DG model is five times as efficient as a standard DG method in simulating dynamics on domains of very large spatial extension. In absorbing layer tests, the extended DG model yields solutions of competitive accuracy in the finite region of interest.
We introduce an extended discontinuous Galerkin discretization of hyperbolic–parabolic problems on multidimensional semiinfinite domains. Building on previous work on the onedimensional case, we split the stripshaped computational domain into a bounded region, discretized by means of discontinuous finite elements using Legendre basis functions, and an unbounded subdomain, where scaled Laguerre functions are used as a basis. Numerical fluxes at the interface allow for a seamless coupling of the two regions. The resulting coupling strategy is shown to produce accurate numerical solutions in tests on both linear and nonlinear scalar and vectorial model problems. In addition, an efficient absorbing layer can be simulated in the semiinfinite part of the domain in order to damp outgoing signals with negligible spurious reflections at the interface. By tuning the scaling parameter of the Laguerre basis functions, the extended DG scheme simulates transient dynamics over large spatial scales with a substantial reduction in computational cost at a given accuracy level compared to standard singledomain discontinuous finite element techniques.
We develop a novel Moving Particle Simulation (MPS) method to calculate the motion of fibers floating in sheared liquids accurately. To introduce the rotational degrees of freedom into MPS particles, we employ the micropolar fluid model. The proposed method can reproduce the periodic rotational motion of a fiber predicted by the Jeffery's theory, which cannot be captured by the conventional MPS.
We develop a novel Moving Particle Simulation (MPS) method to reproduce the motion of fibers floating in sheared liquids accurately. In conventional MPS schemes, if a fiber suspended in a liquid is represented by a onedimensional array of MPS particles, it is entirely aligned to the flow direction due to the lack of shear stress difference between fiber–liquid interfaces. To address this problem, we employ the micropolar fluid model to introduce rotational degrees of freedom into the MPS particles. The translational motion of liquid and solid particles and the rotation of solid particles are calculated with the explicit MPS algorithm. The fiber is modeled as an array of micropolar fluid particles bonded with stretching, bending and torsional potentials. The motion of a single rigid fiber is simulated in a threedimensional shear flow generated between two moving solid walls. We show that the proposed method is capable of reproducing the fiber motion predicted by Jeffery's theory which is different from the conventional MPS simulations.
A novel physicsbased preconditioner of the Jacobianfree Newton–Krylov approach is developed to solve the Navier–Stokes equation using the NIM. The proposed preconditioner leads to huge reduction in condition number by clustering of eigenvalues. Therefore, GMRES convergence improves which drastically reduces Krylov iterations. The reduction in Krylov iterations saves the CPU runtime substantially.
The nodal integral methods (NIMs) have found widespread use in the nuclear industry for neutron transport problems due to their high accuracy. However, despite considerable development, these methods have limited acceptability among the fluid flow community. One major drawback of these methods is the lack of robust and efficient nonlinear solvers for the algebraic equations resulting from discretization. Since its inception, several modifications have been made to make NIMs more agile, efficient, and accurate. Modified nodal integral method (MNIM) and modified MNIM (M^{2}NIM) are the two most recent and efficient versions of the NIM for fluid flow problems. M^{2}NIM modifies the MNIM by replacing the current time convective velocity with the previous time convective velocity, leading to faster convergence albeit with reduced accuracy. This work proposes a preconditioned Jacobianfree Newton–Krylov approach for solving the Navier–Stokes equation using MNIM. The Krylov solvers do not generally work well without an appropriate preconditioner. Therefore, M^{2}NIM is used here as a preconditioner to accelerate the solution of MNIM. Due to pressure–velocity coupling in the Navier–Stokes equation, developing a quality preconditioner for these equations needs significant effort. The momentum equation is solved using the timesplitting alternate direction implicit method. The velocities obtained from the solution are then used to solve the pressure Poisson equation. The computational results for the Navier–Stokes equation are presented to underscore the advantages of the developed algorithm.
This work introduces an empirical quadraturebased hyperreduction procedure and greedy training algorithm to effectively reduce the computational cost of solving convectiondominated problems with limited training. The reducedorder model is defined as the solution of a residual minimization problem over a nonlinear trial manifold. An onlineefficient method is obtained by using empirical quadrature to approximate the optimality system such that it can be solved with meshindependent operations.
This work introduces an empirical quadraturebased hyperreduction procedure and greedy training algorithm to effectively reduce the computational cost of solving convectiondominated problems with limited training. The proposed approach circumvents the slowly decaying n$$ n $$width limitation of linear model reduction techniques applied to convectiondominated problems by using a nonlinear approximation manifold systematically defined by composing a lowdimensional affine space with bijections of the underlying domain. The reducedorder model is defined as the solution of a residual minimization problem over the nonlinear manifold. An onlineefficient method is obtained by using empirical quadrature to approximate the optimality system such that it can be solved with meshindependent operations. The proposed reducedorder model is trained using a greedy procedure to systematically sample the parameter domain. The effectiveness of the proposed approach is demonstrated on two shockdominated computational fluid dynamics benchmarks.
We use a deep learningbased method to solve the problem of underground seepage without any labeled data. A novel approximationcorrection model is proposed in this paper, which combines neural networks with the asymptotic solution of partial differential equations to construct an asymptotic block. By superimposing the asymptotic block, it can solve problems with unsteady boundary conditions, which greatly enhances the solution accuracy.
Deep learningbased methods for solving partial differential equations have become a research hotspot. The approach builds on the previous work of applying deep learning methods to partial differential equations, which avoid the need for meshing and linearization. However, deep learningbased methods face difficulties in effectively solving complex turbulent systems without using labeled data. Moreover, issues such as failure to converge and unstable solution are frequently encountered. In light of this objective, this paper presents an approximationcorrection model designed for solving the seepage equation featuring unsteady boundaries. The model consists of two neural networks. The first network acts as an asymptotic block, estimating the progression of the solution based on its asymptotic form. The second network serves to finetune any errors identified in the asymptotic block. The solution to the unsteady boundary problem is achieved by superimposing these progressive blocks. In numerical experiments, both a constant flow scenario and a threestage flow scenario in reservoir exploitation are considered. The obtained results show the method's effectiveness when compared to numerical solutions. Furthermore, the error analysis reveals that this method exhibits superior solution accuracy compared to other baseline methods.
1. Simple, efficient and effective troubledcell indicators are developed. 2. Three new adaptive WENO algorithms are proposed. 3. First two algorithms maintain the WENO scheme's accuracy while being less costly. 4. Third algorithm ensures the convergence to entropy solution of the WENO scheme.
Hybrid algorithms are an efficient and popular choice for computing the solutions of hyperbolic conservation laws. In general, hybrid algorithms involve lowcost highorder accurate schemes in smooth regions and nonoscillatory shockcapturing schemes in the vicinity of discontinuities. Troubledcell indicators which measure the smoothness of the solution play a significant role in the efficiency of hybrid algorithms. This article proposes a new troubledcell indicator utilising the smoothness indicators of WENO schemes for hyperbolic conservation laws. The proposed troubledcell indicators are simple, efficient, effective, and are used to construct three new adaptive WENO algorithms of highorder accuracy. The hybrid algorithms developed are independent of the order and type of the WENO reconstruction. For demonstration, we have considered the fifth and seventh order WENOZ reconstruction. The first two algorithms have comparable accuracy and resolution of the solution across discontinuities to that of the WENOZ scheme but at a less computational cost. The third algorithm ensures the convergence of the proposed scheme to the correct entropy solution when applied to a hyperbolic conservation law with nonconvex flux for which the WENO schemes fail. We have performed several 1D and 2D numerical experiments to demonstrate the efficiency of the proposed algorithms and their performance compared with the WENOZ schemes. The proposed algorithms are efficient and take 30%–75% less computational time than the WENOZ schemes while retaining the advantages of WENOZ schemes.
We propose a new framework of multiscale finite element methods with bubble function enrichments and demonstrate their effectiveness through a series of numerical experiments for convectiondominant benchmark problems. As an iterative numerical scheme, the key idea is that the global coarse solution at the current step provides feedback for setting better boundary conditions for constructing bubble functions to improve the accuracy of the solution at the next step until convergence.
We develop a new class of the multiscale finite element method (MsFEM) to solve the convectiondiffusion problems. In the proposed framework, we decompose the solution function space into two parts in MsFEM with locally adaptive bubble function enrichment (LABFE). The first part is the one that the multiscale basis functions can resolve, and the second part is an unresolved part that is taken care of by a set of bubble functions. These bubble functions are defined similarly to construct multiscale basis functions. We exchange the localglobal information through updated local boundary conditions for these bubble functions. The new multiscale solution recovered from the solution of global numerical formulation provides feedback for updating the local boundary conditions on each coarse element. As the approach iterates, the quality of MsFEMLABFE solutions improves since these multiscale basis functions with bubble function enrichment are expected to capture the multiscale feature of the approximate solution more accurately. However, the most expansive part of the algorithm is reconstructing the bubble functions on each coarse element. To reduce the overhead of the bubble function reconstruction, we update the local bubble function only when the intermediate multiscale solution is not well resolved within the region corresponding to the sharp local gradient or the discontinuity of the solution. We illustrate the effectiveness of the proposed method through some numerical examples for convectiondiffusion benchmark problems.
In this paper, a unified gas kinetic expression form that can describe the twodimensional planar nozzle flow and the axisymmetric nozzle flow has been established based on previous studies, and the model equations have been solved uniformly by the gaskinetic unified algorithm (GKUA) for rarefied transition to continuum flows. The presented simulated results from the test cases show promising of simulating gas flow from transitional flow to continuum flow in the same flow domain, especially for those flow involving lowspeed gas flow in rarefied regime which is otherwise practically difficult using direct simulation Monte Carlo (DSMC).
For highaltitude nozzle or micronozzle flows and other gas flows in the slip or transition flow regime, how to solve it reliably has always been a difficult problem. In this paper, a unified gas kinetic expression is presented to describe the twodimensional planar and the axisymmetric nozzle flows, and the computable modeling of the Boltzmann equation is developed at the first time for the nozzle flows by introducing the gas molecular collision relaxing parameter and the local equilibrium distribution function integrated in the unified expression with the flow state controlling parameter, including the macroscopic flow variables, the gas viscosity transport coefficient, the thermodynamic effect, the molecular power law, and molecular models, covering a full spectrum of flow regimes. The boundary conditions involved in the nozzle flow have been mathematically expressed at the level of the gas molecular velocity distribution function, and the model equations have been solved uniformly by the gaskinetic unified algorithm (GKUA) for rarefied transition to continuum flows. The presented simulated results from the test cases show promising of simulating gas flow from transitional flow to continuum flow in the same flow domain, especially for those flow involving lowspeed gas flow in rarefied regime, which is otherwise practically difficult using direct simulation Monte Carlo (DSMC).
By relaxing the interlayer surface, a hyperbolic system with analytical eigenstructure is constructed to facilitate the programming and the scheme development. The major contribution of the current study is proposing a novel discretization method for nonconservative and source terms to guarantee the stationary steady states of water and the uniform concentration in denser layer, especially at wetdry fronts
This study considers a watersolute mixture underflow coupled with ambient water. Such physical phenomenons are modelled by a twolayer stratified SaintVenant system with a scalar transport equation in this paper. The governing system is first relaxed to a hyperbolic system discretized by a pathconservative algorithm. In such framework, a novel discrete formula of nonconservative and source terms, is proposed to be able to guarantee the wellbalanced property for the hydrodynamic model and preserve the uniform concentration in underflows, especially at wetdry fronts. The performances of the developed numerical schemes are verified by several tests.
Publication date: 1 November 2023
Source: Journal of Computational Physics, Volume 492
Author(s): Fan Zhang, Jian Cheng
Publication date: 1 November 2023
Source: Journal of Computational Physics, Volume 492
Author(s): Xiaoyu Wei, Andreas Klöckner, Robert C. Kirby
Publication date: 1 November 2023
Source: Journal of Computational Physics, Volume 492
Author(s): QiZhi He, Mauro Perego, Amanda A. Howard, George Em Karniadakis, Panos Stinis
Publication date: 1 November 2023
Source: Journal of Computational Physics, Volume 492
Author(s): Hongtao Liu, Xiaofeng Cai, Yong Cao, Giovanni Lapenta
Publication date: 1 November 2023
Source: Journal of Computational Physics, Volume 492
Author(s): K. Chung, F. Fei, M.H. Gorji, P. Jenny
Publication date: 1 November 2023
Source: Journal of Computational Physics, Volume 492
Author(s): Fabian Laakmann, Kaibo Hu, Patrick E. Farrell
Publication date: 1 November 2023
Source: Journal of Computational Physics, Volume 492
Author(s): Vidhi Zala, Akil Narayan, Robert M. Kirby
Publication date: 1 November 2023
Source: Journal of Computational Physics, Volume 492
Author(s): Somayajulu L.N. Dhulipala, Yifeng Che, Michael D. Shields
Publication date: 1 November 2023
Source: Journal of Computational Physics, Volume 492
Author(s): Meng Zhao, Lijian Jiang
Publication date: 1 November 2023
Source: Journal of Computational Physics, Volume 492
Author(s): Nicolás A. Barnafi, Luca F. Pavarino, Simone Scacchi
We propose a technique for performing spectral (in time) analysis of spatiallyresolved flowfield data, without needing any temporal resolution or information. This is achieved by combining projectionbased reducedorder modeling with spectral proper orthogonal decomposition. In this method, spaceonly proper orthogonal decomposition is first performed on velocity data to identify a subspace onto which the known equations of motion are projected, following standard Galerkin projection techniques. The resulting reducedorder model is then utilized to generate timeresolved trajectories of data. Spectral proper orthogonal decomposition (SPOD) is then applied to this modelgenerated data to obtain a prediction of the spectral content of the system, while predicted SPOD modes can be obtained by lifting back to the original velocity field domain. This method is first demonstrated on a forced, randomly generated linear system, before being applied to study and reconstruct the spectral content of twodimensional flow over two collinear flat plates perpendicular to an oncoming flow. At the range of Reynolds numbers considered, this configuration features an unsteady wake characterized by the formation and interaction of vortical structures in the wake. Depending on the Reynolds number, the wake can be periodic or feature broadband behavior, making it an insightful test case to assess the performance of the proposed method. In particular, we show that this method can accurately recover the spectral content of periodic, quasiperiodic, and broadband flows without utilizing any temporal information in the original data. To emphasize that temporal resolution is not required, we show that the predictive accuracy of the proposed method is robust to using temporallysubsampled data.
Wing–gust encounters cause harmful lift transients that can be mitigated through maneuvering of the wing. This paper presents a method to generate an openloop (i.e., prescribed) maneuver that optimally regulates the lift on the wing during a transverse gust encounter. Obtaining an optimal maneuver is important for laboratory experiments on the physics of wing–gust interactions and may be useful for the future design of feedback controllers. Prior work of the authors has shown that an Iterative Maneuver Optimization (IMO) framework can generate an optimal maneuver by using a surrogate model to propose a control signal that is then tested in experiment or highfidelity simulation. The input to the surrogate model is updated to account for differences between the test data and the expected output. The optimal maneuver is obtained through iteration of this process. This paper simplifies the IMO method by replacing the surrogate model with the classical lift model of Theodorsen, removing the process of optimization over the surrogate model, and removing the requirement to know the timeaveraged profile of the gust. The proposed method, referred to as Simplified IMO (SIMO), only requires input and output data collected from simulations or experiments that interact with the gust. Numerical simulations using a Leading Edge Suction Parameter modulated Discrete Vortex Model are presented to generate the input and output data of the wing–gust encounters for this paper. Experiments in a towing tank also validated the SIMO method. The results show an optimal pitch maneuver and an optimal plunge maneuver that can each regulate lift during a transverse gust encounter.
Perching and hovering are two bioinspired flight maneuvers that have relevance in engineering, especially for smallscale uncrewed air vehicles. In a perching maneuver, the vehicle decelerates to zero velocity while pitching or plunging, and in hovering the pitch and plunge motion kinematics are used to generate fluid dynamic forces even when the vehicle velocity is zero. Even for an airfoil, the fluid dynamics of such maneuvers pose challenges for loworder modeling because of the timevarying freestream velocity, high amplitudes and rates of the motion kinematics, intermittent formation and shedding of the leadingedge vortex (LEV), and the strong effects of the shed vorticity on the loads. In an earlier work by the authors, a leadingedge suction parameter (LESP) was developed to predict intermittent LEV formation for roundleadingedge airfoils undergoing arbitrary variation in pitch and plunge at a constant freestream velocity. In this research, the LESP criterion is extended to situations where the freestream velocity is varying or zero. A discrete vortex method based on this criterion is developed and the results are compared against those from a computational fluid dynamics (CFD) method. Abstractions of perching and hovering maneuvers are used to validate the predictions in highly unsteady vortexdominated flows, where the timevarying freestream/translational velocity is small in magnitude compared to other contributions to the velocity experienced by the airfoil. Time instants of LEV formation, flow features, and force coefficient histories for the various motion kinematics from the method and CFD are obtained and compared. The LESP criterion is seen to be successful in predicting the start of LEV formation, and the discrete vortex method is effective in modeling the flow development and forces on the airfoil.
Both schooling behavior and burstandcoast gait could improve fish swimming performance. The extent to which fish can improve their swimming performance by combining these two strategies is still unknown. By examining two selfpropelled pitching foils positioned sidebyside at different duty cycles (DC), we examine swimming speed and cost of transport efficiency (CoT) using the opensource immersed boundary software IBAMR. We find that a stable schooling formation can only be maintained if both foils employ similar and moderate DC values. In these cases, vortex interactions increase foils’ lateral movements, but not their swimming speed or efficiency. Additionally, we examine vortex interactions in both “schooling" and “fission" scenarios (which are determined by DC). The research provides useful insights into fish behavior and valuable information for designing bioinspired underwater robots.
The influence of an inviscid planar wall on the temporal development of the longwavelength instability of a trailing vortex pair is formulated analytically and studied numerically. The center positions and deformation perturbations of the trailing vortices are marched forward in time via the vortex filament method based on Biot–Savart induction. An optimal perturbation analysis of the vortex system determines the wavenumber and initial condition that yield maximum perturbation growth for any instant in time. Direct integration of the vortex system highlights its sensitivity to initial conditions and the time dependence of the optimal wavenumber, which are not features of the classical free vortex pair. As the counterrotating vortex pair approaches the wall, the wavenumber for maximum growth shifts to a higher value than what is predicted for the Crow instability of vortices in an unbounded fluid. The present analysis demonstrates that the local suppression of the Crow instability near a planar wall may be described without recourse to viscous fluid arguments.
This paper surveys machinelearningbased superresolution reconstruction for vortical flows. Super resolution aims to find the highresolution flow fields from lowresolution data and is generally an approach used in image reconstruction. In addition to surveying a variety of recent superresolution applications, we provide case studies of superresolution analysis for an example of twodimensional decaying isotropic turbulence. We demonstrate that physicsinspired model designs enable successful reconstruction of vortical flows from spatially limited measurements. We also discuss the challenges and outlooks of machinelearningbased superresolution analysis for fluid flow applications. The insights gained from this study can be leveraged for superresolution analysis of numerical and experimental flow data.
Passive flow control is commonly used on bluff bodies for drag and oscillating lift reduction across a range of engineering applications. This research explores a spanwise undulated cylinder inspired by seal whiskers that is shown to reduce hydrodynamic forces when compared to smooth cylinders. Although the fluid flow over this complex geometry has been documented experimentally and computationally, investigations surrounding geometric modifications to the undulation topography have been limited, and fluid mechanisms by which force reduction is induced have not been fully examined. Five variations of undulation wavelength are simulated at Reynolds number \(\text {Re}=250\) and compared with results from a smooth elliptical cylinder. Vortex structures and turbulence kinetic energy (TKE) transfer in the wake are analyzed to explain how undulation wavelength affects force reduction. Modifications to the undulation wavelength generate a variety of flow patterns including alternating vortex rollers and hairpin vortices. Maximum force reduction is observed at wavelengths that are large enough to allow hairpin vortices to develop without intersecting each other and small enough to prevent the generation of additional alternating flow structures. The differences in flow structures modify the magnitude and location of TKE production and dissipation due to changes in mean and fluctuating strain. Decreased TKE production and increased dissipation in the near wake result in overall lower TKE and force reduction. Understanding the flow physics linking geometry to force reduction will guide appropriate parameter selection in bioinspired design applications.
Largeamplitude flow disturbances, or gusts, can drastically alter the aerodynamic forces on an airfoil and are regularly investigated through wind tunnel (or water tunnel) experiments. The gusts generated in those experiments are often further analyzed using numerical simulations, but usually without fully accounting for the wind tunnel walls or gust generator. The current work investigates the wind tunnel effects on the predicted lift response and flow field using a computational framework that models the viscous flow around the airfoil but treats the tunnel walls and gust generation as inviscid boundary conditions. We apply this model to three examples and compare the predicted gust response with the responses predicted by a freespace viscous model and a classical unsteady aerodynamics model to highlight the wind tunnel effects. We find that the wind tunnel modeling introduces nonnegligible effects depending on the airfoil and gust configurations. These effects include the confinement effect of the wind tunnel walls and the triggering of flow separation when it does not occur in the corresponding freespace model. In the last example, we also note that this virtual counterpart of an actual wind tunnel can be paired with experiments through data assimilation to increase the accuracy of the gust response or perform parameter estimation.
In this study, shock standoff distances for thermally and chemically nonequilibrium flows of nitrogen over wedges are computationally investigated via a hypersonic computational fluid dynamics solver, hyperReactingFoam by spanning a parameter space that consists of ranges of Mach number, 4–10, specific heat ratio, 1.40–1.61 and wedge angles, 60 \(^\circ \) –90 \(^\circ \) . Then, the space is reduced into the parameters of inverse density ratio across the shock and dimensionless wedge angle which will be used as variables for quadratic functions that represent shock standoff distances. Besides the functions of shock standoff distances, detached shock profiles of computationally modeled flows are represented by parabolic equations. The flows are observed to be chemically frozen for Mach number ranges of 4–5 regardless of the specific heat ratio value of the nitrogen mixture. Our results show that the shock standoff distance decreases as Mach number is increased from 4 to 7, if the wedge angle and freestream specific heat ratio are kept the same. On the other hand, if Mach number is increased beyond 7, the shock standoff distance starts to extend due to the dissociation of nitrogen molecules behind the shock wave. At Mach 10, nitrogen completely dissociates over 90 \(^\circ \) wedge for all specific heat ratios considered in the present study. Increased leading edge angle of the wedge or specific heat ratio of freestream yields longer shock standoff distance.
The linear stability of a compressible flow in a pipe is examined using a modal analysis. A steady fully developed flow of a calorifically perfect gas, driven by a constant body acceleration, in a pipe of circular cross section is perturbed by smallamplitude normal modes and the temporal stability of the system is studied. In contrast to the incompressible pipe flow that is linearly stable for all modal perturbations, the compressible flow is unstable at finite Mach numbers due to modes that do not have a counterpart in the incompressible limit. We obtain these higher modes for a pipe flow through numerical solution of the stability equations. The higher modes are distinguished into an “odd” and an “even” family based on the variation of their wavespeeds with wavenumber. The classical theorems of stability are extended to cylindrical coordinates and are used to obtain the critical Mach numbers below which the higher modes are always stable. The critical Reynolds number is calculated as a function of Mach number for the even family of modes, which are the least stable at finite Mach numbers. The numerical solution of the stability equations in the high Reynolds number limit demonstrates that viscosity is essential for destabilizing the even family of modes. An asymptotic analysis is carried out at high Reynolds numbers to obtain the scalings, and solutions for the eigenvalues in the high Reynolds number limit for the lower and upper branches of the stability curve.