Compressible Viscous Flow Research Articles

Linstab2D: stability and resolvent analysis of compressible viscous flows in MATLAB

Linstab2D: stability and resolvent analysis of compressible viscous flows in MATLAB

Multirate time-integration based on dynamic ODE partitioning through adaptively refined meshes for compressible fluid dynamics

In this paper, we apply the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et al. [1,2] to dynamically partitioned systems arising from adaptive mesh refinement. The P-ERK schemes enable multirate time-integration with no changes in the spatial discretization methodology, making them readily implementable in existing codes that employ a method-of-lines approach.We show that speedup compared to a range of state of the art Runge-Kutta methods can be realized, despite additional overhead due to the dynamic re-assignment of flagging variables and restricting nonlinear stability properties. The effectiveness of the approach is demonstrated for a range of simulation setups for viscous and inviscid convection-dominated compressible flows for which we provide a reproducibility repository.In addition, we perform a thorough investigation of the nonlinear stability properties of the Paired-Explicit Runge-Kutta schemes regarding limitations due to the violation of monotonicity properties of the underlying spatial discretization. Furthermore, we present a novel approach for estimating the relevant eigenvalues of large Jacobians required for the optimization of stability polynomials.

On steady state of viscous compressible heat conducting full magnetohydrodynamic equations

This paper is concerned with the study of equations of viscous compressible and heat-conducting full magnetohydrodynamic (MHD) steady flows in a horizontal layer under the gravitational force and a large temperature gradient across the layer. We assume as boundary conditions, periodic conditions in the horizontal directions, while in the vertical directions, slip-boundary is assumed for the velocity, vertical conditions for the magnetic field, and fixed temperature or fixed heat flux are prescribed for the temperature. The existence of stationary solution in a small neighborhood of a stationary profile close to hydrostatic state is obtained in Sobolev spaces as a fixed point of some nonlinear operator.

Double distribution function-based lattice Boltzmann flux solver for simulation of compressible viscous flows

In this work, a double distribution function-based lattice Boltzmann flux solver (LBFS) is proposed for simulating compressible viscous flows. This approach utilizes the double distribution function compressible lattice Boltzmann model and employs Chapman–Enskog expansion analysis to connect the lattice Boltzmann equation (LBE) with the Navier–Stokes (N–S) equations. Unlike conventional computational fluid dynamics methods that compute inviscid and viscous fluxes separately, the present method simultaneously evaluates both types of fluxes at the cell interface by locally reconstructing the solution of the LBE. Recognizing the significance of considering the non-equilibrium part of distribution functions for viscous flows, a straightforward method is introduced to calculate this component. This facilitates the derivation of computational expressions for macroscopic conservative variables and fluxes in the N–S equations. To validate the accuracy and stability of the present numerical scheme, various benchmark problems, including shock tube problem, Couette flow, lid-driven cavity flow, and flow around the NACA0012 airfoil, are tested. The obtained numerical results are compared with analytical solutions or existing reference data, confirming the capability of the proposed LBFS to deliver accurate and stable numerical results for compressible flows. Moreover, this method demonstrates effectiveness in handling viscous flow problems on non-uniform grids and with curved boundaries.

Rigorous justification of the hydrostatic approximation limit of viscous compressible flows

This paper considers the asymptotic limit of small aspect ratio between vertical and horizontal spatial scales for viscous isothermal compressible flows. In particular, it is observed that fast vertical acoustic waves arise and induce an averaging mechanism of the density in the vertical variable, which at the limit leads to the hydrostatic approximation of compressible flows, i.e., the compressible primitive equations of atmospheric dynamics. We justify the hydrostatic approximation for general as well as “well-prepared” initial data. The initial data is called well-prepared when it is close to the hydrostatic balance in a strong topology. Moreover, the convergence rate is calculated in the well-prepared initial data case in terms of the aspect ratio, as the latter goes to zero.

Earthquake Cycle Mechanics During Caldera Collapse: Simulating the 2018 Kı̄lauea Eruption

AbstractIn multiple observed caldera‐forming eruptions, the rock overlying a draining magma reservoir dropped downward along ring faults in sequences of discrete collapse earthquakes. These sequences are analogous to tectonic earthquake cycles and provide opportunities to examine fault mechanics and collapse eruption dynamics over multiple events. Collapse earthquake cycles have been studied with zero‐dimensional slider‐block models, but these do not account for the complicated interplay between fluid and elastic dynamics or for factors such as the heterogeneous fault properties and non‐vertical ring fault geometries often inferred at volcanoes. We present two‐dimensional axisymmetric mafic piston‐like collapse earthquake cycle models that include rate‐and‐state friction, fully‐dynamic elasticity, and compressible viscous fluid magma flow. We demonstrate that collapse earthquake intervals and magnitudes are highly sensitive to inertial effects, evolving stress fields, fault geometry, and depth‐varying fault friction. Given the consistent earthquake cycles observed in most eruptions, this suggests that ring faults can quickly stabilize and often become nearly vertical at depth. We use the well‐monitored 2018 collapse sequence at Kı̄lauea as a case study. Our model can produce many features of Kı̄lauea seismic and geodetic observations, except for a significant amount of interseismic slip, which cannot be readily explained with simple rate‐and‐state friction parameterizations.

Viscous compressible flow from a semi-sealed porous system: Mechanisms and wave-mediated diffusion

Viscous compressible flow from a semi-sealed porous system: Mechanisms and wave-mediated diffusion

An Eulerian–Lagrangian particle method for weakly compressible viscous flows using peridynamic differential operator

This study proposes a peridynamic differential operator (PDDO)-based Eulerian–Lagrangian hybrid particle method for weakly compressible viscous flows. The PDDO is utilized to transform the governing partial differential equations into their integral form; hence, the issue of local non-differentiability is eliminated. Both the Eulerian and Lagrangian formulations of particle methods for solving the governing integral equations are derived and then combined to propose the hybrid method. In the proposed method, the Eulerian and Lagrangian formulation is utilized to solve the governing equations in the inner computational domain and in areas with free surfaces, respectively. The results of these two domains are then merged to arrive at the solution. In such a way, the hybrid particle method effectively balances the computational demands and applicability. The numerical stability and interpolation consistency of the proposed method are shown by introducing the laminar viscosity model. A detailed numerical procedure is provided involving the boundary conditions and time-stepping strategy. The proposed method is validated by several benchmark problems. Furthermore, the method was applied to several benchmark problems including the hydrostatic test, Taylor–Green vortex, and numerical wave generation. The results of these numerical examples suggest that the proposed method is computationally less demanding while maintaining accuracy.

New similarity laws reduced from local Mach factors in longitudinal–transverse force theory

The rapid advancement in aeronautics has led to the emergence of intricate dynamic processes and structures, such as vortices, shock waves, flow separation, and turbulence, resulting from the flow around airfoils. Acquiring a profound understanding of these local structures and unraveling the physical mechanisms underlying flow phenomena represents an essential and challenging issue in the field of flight science. In this research, the longitudinal–transverse force theory (L–T force theory), as proposed by previous researchers, is employed to quantitatively assess contributions of local flow structures to aerodynamic forces. Specifically, the research encompasses an analysis of steady and viscous compressible flow over the Royal Aircraft Establishment (RAE)-2822 airfoil, with free-stream Mach number (M∞) ranging from 0.1 to 2.0. We comprehensively estimate longitudinal forces (L-force) and transverse forces (T-force), along with effects of compressibility on aerodynamic forces. Furthermore, recognizing the necessity for high-precision algorithms in the computation of L–T force theory, this investigation utilizes a sixth-order accuracy algorithm for spatial discretization and differencing. Our analysis reveals that the influence of compressibility and the contributions of L-forces to aerodynamic forces become increasingly significant in high M∞ regimes as shearing processes weaken. Additionally, a new similarity law is established to characterize aerodynamic forces acting on the RAE-2822 airfoil based on a novel moderating factor, ζmo, reduced from local Mach factors in the L–T force theory. This coefficient, ζmo, elucidates the degree to which transverse processes are modulated by longitudinal processes. Various angles of attack α and airfoils have also been analyzed, including National Advisory Committee for Aeronautics (NACA)0012 and NACA0006, by introducing a parameter denoted as κ to further validate the universality of the new similarity laws. The results demonstrate a high degree of accuracy in fitting the aerodynamic coefficients.

High‐order gas kinetic flux solver for viscous compressible flow simulations

AbstractAlthough the gas kinetic schemes (GKS) have emerged as one of the powerful tools for simulating compressible flows, they exhibit several shortcomings. Since the local solution of continuous Boltzmann equation with the Maxwellian distribution function is used to calculate the numerical fluxes at the cell interface, the flux expression in GKS is usually more complicated. In this paper, a high‐order simplified gas kinetic flux solver (GKFS) is presented for simulating two‐dimensional compressible flows. Circular function‐based GKFS (C‐GKFS), which simplifies the Maxwellian distribution function into the circular function, combined with an improved weighted essentially non‐oscillatory (WENO‐Z) scheme is applied to capture more details of the flow fields with fewer grids. As a result, a simple high‐order accurate C‐GKFS is obtained, which improves the computing efficiency and reduce its complexity to facilitate the practical application of engineering. A series of benchmark‐test problems are simulated and good agreement can be obtained compared with the references, which demonstrate that the high‐order C‐GKFS can achieve the desired accuracy.

A New Fifth-Order Weighted Compact Nonlinear Scheme with Multi-Order Candidates Weighting for Hyperbolic Conservation Laws

ABSTRACT Based on one 5-point and two 2-point stencils, a nonlinear weighted scheme adaptive to the weighting of multi-order interpolation candidates for shock capturing is proposed. The JS weight function is adopted. The smoothness indicator of the 5-point stencil is specially designed with a higher-order leading term similar to the reference global smoothness indicator τ in the Z weighting. The design maintains that the nonlinear weights satisfy the sufficient condition for the scheme to avoid degradation at extreme points. The form of the weighting is also simplified, obviously reducing the weighting procedure's computational costs. Inviscid and viscous compressible flow problems are simulated to test the performance of the newly designed nonlinear weights based on the weighted compact nonlinear scheme. The numerical results show no obvious oscillations near the discontinuity, and the resolution of both discontinuities and smooth regions is better than JS, Z, and ZQ weighting.

Dissipation in nonequilibrium thermodynamics and its connection to the Rayleighian functional

We examine quantitatively the role of dissipation in nonequilibrium thermodynamics and its connection to variational principles and the Rayleighian functional. The extremum of the Rayleighian is sometimes used to describe the inertialess (dissipation-dominated) dynamics of continuum systems, and it has been applied recently for the modeling of soft matter dynamics. We discuss how dissipation is considered within one of the modern complete descriptions of nonequilibrium thermodynamics, namely the single generator bracket formalism. Within this formalism, dissipation is introduced through the use of the dissipation bracket, describing irreversible dynamics, which is added to a Poisson bracket that describes the reversible dynamics of the system. A possible connection with the Rayleighian functional is then demonstrated that in all cases considered herein, the Rayleighian is equal to minus one half of the effective dissipation rate of the Lagrangian functional. The effective dissipation rate is obtained starting with an inertial (i.e., flux-based or velocity-based) system description, involving the Poisson bracket and the primitive part (i.e., without the entropy correction term) of the dissipative bracket. Several examples are discussed in detail, ranging from an algebraic model (damped oscillator) to continuum ones: modeling of fluid flow in porous particle media, viscous Newtonian compressible and incompressible fluid flows, and more interestingly, flow of a nematic liquid-crystalline material.

Исследование влияния химического состава воздушной среды на аэродинамику полета малых летательных аппаратов

The work is devoted to the study of the influence of an air environment with different chemical composition on the aerodynamics of small aircrafts of the airplane type used for reconnaissance purposes in case of fires. The thermodynamic properties of the combustion products formed during the combustion of rubber and foam rubber were calculated. The properties of the medium were taken into account in the mathematical modeling of the flight aerodynamics of an unmanned aerial vehicle. The mathematical model is based on solving conservation equations in a non-isothermal formulation for a turbulent viscous compressible flow using the k-ε turbulence model. The pressure and velocity fields of the environment were obtained during the flow around the wing, and the values of drag and lift forces were compared in the presence of combustion products in the air and without them. In particular, the numerical calculations were carried out in the presence of a vertical temperature gradient in the design area simulating the heating of air masses during fires with the study of the influence of these gradients on the aerodynamic characteristics of the wing. The calculations show that with an increase in the vertical temperature gradient by 60 K/m, the values of lift and drag decrease by 5 %.

Clocking effects on aerodynamics of a gas turbine low pressure axial turbine, part 2: Unsteady simulation

Unsteady, viscous compressible flow is numerically simulated within first two stages of a low pressure turbine of a gas turbine engine under different clocking of its stator blades rows. Zonal DES turbulence modeling has been used for flow simulations. Time-dependent wake flow trajectory is modeled and visualized within the blades passages. All characteristics of the passage wake flow, including its V-shape motion, re-orientation, elongation, expansion, and stretching, are properly simulated. Unsteady flow field results together with those of general aerodynamic performance of the optimum and worst clocking cases are presented and compared with each other. Analyses of the flow simulation results showed that the optimum clocking of the blades rows occurs while the upstream wake flow impinges on the downstream stator blade leading edge region. Final results demonstrated that under the optimum clocking of the second stator, the inlet total pressure of the second rotor blades row and its output power increase by 0.1% and 0.336%, respectively. Consequently, efficiency of the second stage increases by 0.35%. In addition, total output power of the whole two stages increases by 0.237%.

Generalisation of the Penalised Wall Function Method for the Simulation of Turbulent flows With Unfavourable Pressure Gradients

The penalized wall function method for simulation of compressible near-wall turbulent flow regions in the numerical modeling of viscous compressible flows is developed. The method is formulated as a differential condition to match the outer and the wall function solutions and is based on a generalized characteristic-based volume penalization method to transfer shear stress from the outer region of the boundary layer to the wall. The method is modified to extend its applicability to turbulent flows with adverse pressure gradient, when separation and reattachment zones are formed, as well as to use computational meshes with coarser near-wall resolution. These advantages are demonstrated for two test problems, namely, the flow over a flat plate with zero and adverse pressure gradients.

Quadrature-free forms of discontinuous Galerkin methods in solving compressible flows on triangular and tetrahedral grids

The nonlinear stability would affect the accuracy and robustness of the discontinuous Galerkin (DG) method in solving the inviscid and viscous compressible flows. This issue is studied by proposing a new framework to incorporate the quadrature-based and quadrature-free discontinuous Galerkin methods on triangular and tetrahedral grids. Four nodal DG schemes are derived by choosing special test functions and different collocation points for the approximate solution and flux function. The new framework can be more efficient in computational cost and easier in code implementation especially for the quadrature-free scheme. These four schemes are the same for the linear system but different for the nonlinear system due to aliasing errors. Numerical examples are provided to validate the performance in accuracy and nonlinear stability.

BE-BDF2 Time Integration Scheme Equipped with Richardson Extrapolation for Unsteady Compressible Flows

In this work we investigate the effectiveness of the Backward Euler-Backward Differentiation Formula (BE-BDF2) in solving unsteady compressible inviscid and viscous flows. Furthermore, to improve its accuracy and its order of convergence, we have equipped this time integration method with the Richardson Extrapolation (RE) technique. The BE-BDF2 scheme is a second-order accurate, A-stable, L-stable and self-starting scheme. It has two stages: the first one is the simple Backward Euler (BE) and the second one is a second-order Backward Differentiation Formula (BDF2) that uses an intermediate and a past solution. The RE is a very simple and powerful technique that can be used to increase the order of accuracy of any approximation process by eliminating the lowest order error term(s) from its asymptotic error expansion. The spatial approximation of the governing Navier–Stokes equations is performed with a high-order accurate discontinuous Galerkin (dG) method. The presented numerical results for canonical test cases, i.e., the isentropic convecting vortex and the unsteady vortex shedding behind a circular cylinder, aim to assess the performance of the BE-BDF2 scheme, in its standard version and equipped with RE, by comparing it with the ones obtained by using more classical methods, like the BDF2, the second-order accurate Crank–Nicolson (CN2) and the explicit third-order accurate Strong Stability Preserving Runge–Kutta scheme (SSP-RK3).

Application of arbitrary Lagrangian–Eulerian unstructured finite volume lattice Boltzmann method to simulate compressible viscous flows over moving bodies

Application of arbitrary Lagrangian–Eulerian unstructured finite volume lattice Boltzmann method to simulate compressible viscous flows over moving bodies

Computational Studies of the Aerophysical Characteristics on the Head Part of the Supersonic Body of Rotation During Flight Along Trajectories

The use of modern computer systems for calculating the aerophysical characteristics of supersonic bodies of revolution together with the calculation of the flight trajectory under conditions of continuously changing Mach and Reynolds numbers of the oncoming flow, the presence of transient regimes and non-isothermality in near-wall flows is often not possible due to the complexity of the organisation of the computational process. By using a number of integral methods for calculating inviscid flows and viscous compressible near-wall flows, taking into account the non isothermality and intermittency of the boundary layer, a method has been developed for calculating the friction and heating of supersonic bodies of revolution on the flight trajectory. On the basis of well-known flight experiments, the correlation dependences of the Reynolds numbers of the beginning of the transition on sharp cones were obtained and the method of corrections for spherical blunting of the head part was implemented. By calculating the intermittency function, the changeability of flow regimes in the boundary layer is established for a number of trajectories of the supersonic uncontrolled rocket projectile. Calculated dependences for aerodynamic friction, heat fluxes and projectile wall temperature on the flight time are obtained. The concept of combined calculation of aerophysical characteristics on the flight trajectory using numerical and integral methods is proposed.

Assessment of volume penalization and immersed boundary methods for compressible flows with various thermal boundary conditions

A continuous forcing immersed boundary method (IBM) and a volume penalization method are investigated to simulate compressible viscous flows past an isothermal or adiabatic solid obstacle using a body non-conformal Cartesian grid. The present methods are validated on a large range of flow configurations such as incompressible and compressible flows (steady or unsteady) past isothermal or adiabatic solid obstacles. An adiabatic condition with implicit treatment of the source term is derived for the penalization method, taking advantage of the framework of the ghost-cell methods. Accuracy and computational cost of the methods are discussed. Three-dimensional supersonic flow configurations are carried out with the penalization method. The coupling between the porosity parameter and the different boundary conditions of the present penalization method is investigated in order to properly reproduce the wave reflection on a solid wall. Finally, a new penalization model is designed to improve the shock wave reflection on a solid surface while significantly reducing the computational cost compared to the previous porosity model.