Compressible vortex rings, usually formed at the open end of a shock tube, often show interesting phenomena during their formation, evolution, and propagation depending on the shock Mach number (Ms) and exit flow conditions. The Mach number of the translating compressible vortex rings (Mv) investigated so far in the literature is subsonic as, the shock tube pressure ratio (PR) considered is relatively low. In this numerical study we focus on low to high vortex ring Mach numbers (0.31 < Mv < 1.08) cases with a particular focus on very high Mv cases that are not been reported in experiments as, it is difficult to obtain in laboratory. Using hydrogen as a driver section gas inside the shock tube, a supersonic compressible vortex ring (Mv > 1) is obtained for first time. It is established that the SST k-ω based DES turbulent model replicates the experimental observation better than the previously published results at different stages of development of the vortex ring. DES, which is an inbuilt hybrid of LES and RANS approaches is evoked that can automatically switch to the sub-grid scale (SGS) model in the LES regions (i.e. with different scale vortical structures) and to a RANS model in the rest of the region (i.e. where the grid spacing is greater than the turbulent length scale). The DES model can predict characteristics of the shear layer vortices as well as counter-rotating vortex rings (CRVRs) as reported in the experimental measurements. Formation of multiple triple points and the corresponding slip-stream shear layers and thus multiple CRVRs behind the primary vortex ring at different radial locations, in addition to the usual CRVRs, appears to be a unique characteristic for high Mach number vortex rings. For high PR, H2, case during formation stage, a vortex layer of reverse circulation (that of primary vortex ring) is formed (contd...)
We study immiscible two-phase flow of a compressible and an incompressible fluid inside a capillary tube of varying radius under steady-state conditions. The incompressible fluid is Newtonian and the compressible fluid is an inviscid ideal gas. The surface tension associated with the interfaces between the two fluids introduces capillary forces that vary along the tube due to the variation in the tube radius. The interplay between effects due to the capillary forces and the compressibility results in a set of properties that are different from incompressible two-phase flow. As the fluids move …
We propose a two-population lattice Boltzmann model on standard lattices for the simulation of compressible flows. The model is fully on-lattice and uses the single relaxation time Bhatnagar-Gross-Krook kinetic equations along with appropriate correction terms to recover the Navier-Stokes-Fourier equations. The accuracy and performance of the model are analyzed through simulations of compressible benchmark cases including Sod shock tube, sound generation in shock-vortex interaction and compressible decaying turbulence in a box with eddy shocklets. It is demonstrated that the present model prov…
As a first step towards the asymptotic description of coherent structures in compressible shear flows, we present a description of nonlinear equilibrium solutions of the Navier--Stokes equations in the compressible asymptotic suction boundary layer (ASBL). The free-stream Mach number is assumed to be $< 0.8$ so that the flow is in the subsonic regime and we assume a perfect gas. We extend the large-Reynolds number free-stream coherent structure theory of \cite{deguchi_hall_2014a} for incompressible ASBL flow to describe a nonlinear interaction in a thin layer situated just below the free-strea…
In our recent works we proposed a theory of turbulence in inertial gas flow via the mean field effect of an intermolecular potential. We found that, in inertial flow, turbulence indeed spontaneously develops from a laminar initial condition, just as observed in nature and experiments. However, we also found that density and temperature in our inertial flow model behave unrealistically. The goal of the current work is to demonstrate technical possibility of modeling compressible, turbulent flow at low Mach number where both density and temperature behave in a more realistic fashion. Here we foc…
We investigate the evolution of vortex-surface fields (VSFs) in compressible Taylor--Green flows at Mach numbers ($Ma$) ranging from 0.5 to 2.0 using direct numerical simulation. The formulation of VSFs in incompressible flows is extended to compressible flows, and a mass-based renormalization of VSFs is used to facilitate characterizing the evolution of a particular vortex surface. The effects of the Mach number on the VSF evolution are different in three stages. In the early stage, the jumps of the compressive velocity component near shocklets generate sinks to contract surrounding vortex su…
We present a theory for the three-dimensional evolution of tubes with expandable walls conveying fluid. Our theory can accommodate arbitrary deformations of the tube, arbitrary elasticity of the walls, and both compressible and incompressible flows inside the tube. We also present the theory of propagation of shock waves in such tubes and derive the conservation laws and Rankine-Hugoniot conditions in arbitrary spatial configuration of the tubes, and compute several examples of particular solutions. The theory is derived from a variational treatment of Cosserat rod theory extended to incorpora…
In this work, direct numerical simulations of the compressible fluid equations in turbulent regimes are performed. The behavior of the flow is either dominated by purely turbulent phenomena or by the generation of sound waves in it. Previous studies suggest that three different types of turbulence may happen at the low Mach number limit in polytropic flows: Nearly incompressible (NI), modally equipartitioned compressible (MEC) and compressible wave (CW). The distinction between these types of turbulence is investigated here applying different kinds of forcing. Scaling of density fluctuations w…
The spatial scaling laws of velocity kinetic energy spectrum for compressible turbulence flow and its density-weighted counterpart have been formulated in terms of wavenumber, dissipation rate and Mach number by using dimensional analysis. We have applied the Barenblatt's incomplete similarity theory to both kinetic and density-weighted energy spectrum and showed that, within the initial subrange, both energy spectrums approach the -5/3 power law of the wavenumber, when the Mach number $M$ tends to be naught, unity and infinity, respectively.
This paper is a continuation of our earlier work [Z.L. Guo {\it et al.}, Phys. Rev. E {\bf 88}, 033305 (2013)] where a multiscale numerical scheme based on kinetic model was developed for low speed isothermal flows with arbitrary Knudsen numbers. In this work, a discrete unified gas-kinetic scheme (DUGKS) for compressible flows with the consideration of heat transfer and shock discontinuity is developed based on the Shakhov model with an adjustable Prandtl number. The method is an explicit finite-volume scheme where the transport and collision processes are coupled in the evaluation of the flu…
We perform particle-resolved simulations of subsonic and transonic flows past random arrays of spherical particles. The Reynolds number is held at $Re{\approx}300$ to ensure the flow remains in the continuum regime. At low volume fractions, the drag force increases sharply near a critical Mach number due to the formation of shock waves and reaches a maximum value when the bulk flow is supersonic. Neighbour-induced hydrodynamic interactions reduce the critical Mach number at higher volume fractions. An effective Mach number is introduced to capture the increase in compressibility effects on dra…
A Conservative Time-Accurate Local Time-Stepping DG Scheme Based on a Weakly Compressible Model for Unsteady Low-Mach-Number Flows
Digital Twin Research Center, Institute of Engineering Thermophysics, Chinese Academy of Sciences（中国科学院工程热物理研究所数字孪生研究中心）；Beihang University（北京航空航天大学）；School of Mechanical Engineering, Tianjin University（天津大学机械工程学院）；Rankyee Technology Beijing（北京瑞基科技有限公司）
A family of compressible exact periodic solutions is reported for the first time in an open cavity flow setup. These are found using a novel framework which permits the computation of such solutions in an arbitrary complex geometry. The periodic orbits arise from a synchronised concatenation of convective and acoustic events which strongly depend on the Mach number. This flow-acoustic interaction furnishes the periodic solutions with a remarkable stability and it is found to completely dominate the system's dynamics and the sound directivity. The periodic orbits, which could be called `exact R…
We investigate flows interacting with a square and a fractal shape multi-scale structures in the compressible regime for Mach numbers at subsonic and supersonic upstream conditions using large-eddy-simulations (LES). We also aim at identifying similarities and differences that these interactions have with corresponding interactions in the canonical incompressible flow problem. To account for the geometrical complexity associated with the fractal structures, we apply an immersed boundary method to model the no-slip boundary condition at the solid surfaces, with adequate mesh resolution in the v…
We carry out direct numerical simulation of compressible square duct flow in the range of bulk Mach numbers M_b = 0.2-3, and up to friction Reynolds number Re_τ = 500. The effects of flow compressibility on the secondary motions are found to be negligible, as the typical Mach number associated with the cross-stream flow is always less than 0.1. As in the incompressible case, we find that the wall law for the mean streamwise velocity applies with good approximation with respect to the nearest wall, upon suitable compressibility transformation. The same conclusion also applies to a passive scala…
The direct numerical simulation (DNS) of compressible isotropic turbulence up to the supersonic regime $Ma_{t} = 1.2$ has been investigated by high-order gas-kinetic scheme (HGKS) [{\it{Computers}} \& {\it{Fluids, 192, 2019}}]. In this study, the coarse-grained analysis of subgrid-scale (SGS) turbulent kinetic energy $K_{sgs}$ budget is fully analyzed for constructing one-equation SGS model in the compressible large eddy simulation (LES). The DNS on a much higher turbulent Mach number up to $Ma_{t} = 2.0$ has been obtained by HGKS, which confirms the super robustness of HGKS. Then, the exact c…
Nonmodal transient growth studies and estimation of optimal perturbations have been made for the compressible plane Couette flow with three-dimensional disturbances. The maximum amplification of perturbation energy over time, $G_{\max}$, is found to increase with increasing Reynolds number ${\it Re}$, but decreases with increasing Mach number $M$. More specifically, the optimal energy amplification $G_{\rm opt}$ (the supremum of $G_{\max}$ over both the streamwise and spanwise wavenumbers) is maximum in the incompressible limit and decreases monotonically as $M$ increases. The corresponding op…
Miniature shock tubes are finding growing importance in a variety of interdisciplinary applications. There is a lack of experimental data to validate the existing shock tube flow models that explain the shockwave attenuation in pressure-driven miniature shock tubes. This paper gives insights into the shock formation and shock propagation phenomena in miniature shock tubes of 2mm, 6mm, and 10mm square cross-sections operated at diaphragm rupture pressure ratios in the range 5-25 and driven section initially at ambient conditions. Pressure measurements and visualization studies are carried out i…
Hydrodynamic stability of compressible boundary layers is strongly influenced by Mach number ($M$), Prandtl number ($Pr$) and thermal wall boundary condition. These effects manifest on the flow stability via the flow-thermodynamic interactions. Comprehensive understanding of stability flow physics is of fundamental interest and important for developing predictive tools and closure models for integrated transition-to-turbulence computations. The flow-thermodynamic interactions are examined using linear analysis and direct numerical simulations (DNS) in the following parameter regime: $0.5 \leq …
This work thoroughly investigates a semi-Lagrangian lattice Boltzmann (SLLBM) solver for compressible flows. In contrast to other LBM for compressible flows, the vertices are organized in cells, and interpolation polynomials up to fourth order are used to attain the off-vertex distribution function values. Differing from the recently introduced Particles on Demand (PoD) method, the method operates in a static, non-moving reference frame. Yet the SLLBM in the present formulation grants supersonic flows and exhibits a high degree of Galilean invariance. The SLLBM solver allows for an independent…
A numerical procedure was developed for solving equations for compressible granular multiphase flows in which the particle volume fraction can range dynamically from very dilute to very dense. The procedure uses a low-dissipation and high-order numerical method that can describe shocks and incorporates a particulate model based on kinetic theory. The algorithm separates edges of a computational cell into gas and solid sections where gas- and granular-phase Riemann problems are solved independently. Solutions from these individual Riemann problems are combined to assemble the fully coupled conv…
We study high Mach number limit of the one dimensional piston problem for the full compressible Euler equations of polytropic gas, for both cases that the piston rushes into or recedes from the uniform still gas, at a constant speed. There are two different situations, and one needs to consider measure solutions of the Euler equations to deal with concentration of mass on the piston, or formation of vacuum. We formulate the piston problem in the framework of Radon measure solutions, and show its consistency by proving that the integral weak solutions of the piston problems converge weakly in t…
The HLLC Riemann solver, which resolves both the shock waves and contact discontinuities, is popular to the computational fluid dynamics community studying compressible flow problems with mesh methods. Although it was reported to be used in meshless methods, the crucial information and procedure to realise this scheme within the framework of meshless methods were not clarified fully. Moreover, the capability of the meshless HLLC solver to deal with compressible liquid flows is not completely clear yet as very few related studies have been reported. Therefore, a comprehensive investigation of a…
We revisit the problem of how energy transfer through the turbulent cascade operates in compressible hydrodynamic turbulence. In general, there is no conservative compressible cascade since the kinetic and internal energy reservoirs can exchange energy through pressure dilatation. Moreover, statistically stationary turbulence at high Mach number can only be maintained in nearly isothermal gas, i.e., if excess heat produced by shock compression and kinetic energy dissipation is continuously removed from the system. We mimic this process by a linear cooling term in numerical simulations of turbu…
Most fluid flow problems that are vital in engineering applications involve at least one of the following features: turbulence, shocks, and/or material interfaces. While seemingly different phenomena, these flows all share continuous generation of high wavenumber modes, which we term the '$k_\infty$ irregularity.' In this work, an inviscid regularization technique called 'observable regularization' is proposed for the simulation of two-phase compressible flows. The proposed approach regularizes the equations at the level of the partial differential equation and as a result, any numerical metho…
The validity of the anelastic approximation has recently been questioned in the regime of rapidly-rotating compressible convection in low Prandtl number fluids (Calkins et al. 2015). Given the broad usage and the high computational efficiency of sound-proof approaches in this astrophysically relevant regime, this paper clarifies the conditions for a safe application. The potential of the alternative pseudo-incompressible approximation is investigated, which in contrast to the anelastic approximation is shown to never break down for predicting the point of marginal stability. Its accuracy, howe…
The inverse transfer of magnetic helicity is investigated through direct numerical simulations of large-scale-mechanically-driven turbulent flows in the isothermal ideal magnetohydrodynamics (MHD) framework. The mechanical forcing is either purely solenoidal or purely compressive and the turbulent steady-states considered exhibit root mean square (RMS) Mach numbers 0.1 $\lesssim$ M $\lesssim$ 11. A continuous small-scale electromotive forcing injects magnetic helical fluctuations, which lead to the build-up of ever larger magnetic structures. Spectral scaling exponents are observed which, for …
Supersonic turbulent flows of magnetized gas are believed to play an important role in the dynamics of star-forming clouds in galaxies. Understanding statistical properties of such flows is crucial for developing a theory of star formation. In this letter we propose a unified approach for obtaining the velocity scaling in compressible and super--Alfvénic turbulence, valid for arbitrary sonic Mach number, \ms. We demonstrate with numerical simulations that the scaling can be described with the She--Lévêque formalism, where only one parameter, interpreted as the Hausdorff dimension of the most i…
Fully compressible magnetohydrodynamic (MHD) simulations are a fundamental tool for investigating the role of dynamo amplification in the generation of magnetic fields in deep convective layers of stars. The flows that arise in such environments are characterized by low (sonic) Mach numbers (M_son < 0.01 ). In these regimes, conventional MHD codes typically show excessive dissipation and tend to be inefficient as the Courant-Friedrichs-Lewy (CFL) constraint on the time step becomes too strict. In this work we present a new method for efficiently simulating MHD flows at low Mach numbers in a sp…
The probability density function (PDF) of the gas density in subsonic and supersonic, isothermal, driven turbulence is analyzed with a systematic set of hydrodynamical grid simulations with resolutions up to 1024^3 cells. We performed a series of numerical experiments with root mean square (r.m.s.) Mach number M ranging from the nearly incompressible, subsonic (M=0.1) to the highly compressible, supersonic (M=15) regime. We study the influence of two extreme cases for the driving mechanism by applying a purely solenoidal (divergence-free) and a purely compressive (curl-free) forcing field to d…
We study a singular limit of a scaled compressible Navier--Stokes--Coriolis system driven by both a deterministic and stochastic forcing terms in three dimensions. If the Mach number is comparable to the Froude number with both proportional to say $\varepsilon\ll 1$, whereas the Rossby number scales like $\varepsilon^m$ for $m>1$ large, then we show that any family of weak martingale solution to the $3$-D randomly forced rotating compressible equation (under the influence of a deterministic centrifugal force) converges in probability, as $\varepsilon\rightarrow0$, to the $2$-D incompressible N…
Energy equipartition is a powerful theoretical tool for understanding astrophysical plasmas. It is invoked, for example, to measure magnetic fields in the interstellar medium (ISM), as evidence for small-scale turbulent dynamo action, and, in general, to estimate the energy budget of star-forming molecular clouds. In this study we motivate and explore the role of the volume-averaged root-mean-squared (rms) magnetic coupling term between the turbulent, $δ\mathbf{B}$ and large-scale, $\mathbf{B}_0$ fields, $\left< (δ\mathbf{B}\cdot\mathbf{B}_0)^{2} \right>^{1/2}_{\mathcal{V}}$. By considering th…
