Small Mach Number Research Articles

Weakly Compressible Approximation of the Taylor–Green Vortex Solution

ABSTRACTThe Taylor–Green vortex represents an exact solution of the Navier–Stokes equations in . In this work, an approximation of this solution in two spatial dimensions is proposed for weakly compressible flows. These flows are characterized by small compressibility (or, equivalently, by a small Mach number) and are often employed in computational fluid dynamics to approximate the behaviour of incompressible Newtonian fluids. In this framework, the proposed solution is expected to be a useful benchmark for numerical solvers that implement the weakly compressibility approximation. To this end, some numerical examples are reported in the final section of this work.

How general is the Jensen–Varadhan large deviation functional for 1D conservation laws?

Starting from a microscopic particle model whose hydrodynamic limit under hyperbolic space-time scaling is a 1D conservation law, we derive the large deviation rate function encoding the probability to observe a density profile which is a non entropic shock, and compare this large deviation rate function with the classical Jensen-Varadhan functional, valid for the totally asymmetric exclusion process and the weakly asymmetric exclusion process in the strong asymmetry limit. We find that these two functionals have no reason to coincide, and in this sense Jensen-Varadhan functional is not universal. However, they do coincide in a small Mach number limit, suggesting that universality is restored in this regime. We then compute the leading order correction to the Jensen-Varadhan functional.

Vortex breakdown in the small Mach number regimes

The effect of compressibility for flow inside the cylinder with the top rotating lid (Vogel-Escudier flow) is examined. Three-dimensional Navier–Stokes equations for compressible flow in Cartesian coordinates are used to simulate the flow using open-source OpenFOAM software. The Mach number (Ma) of the flow is varied from 0.1 to 0.3, and the Reynolds number (Re) is varied from 1000 to 5000 for a fixed aspect ratio (Γ = 2.5) of the cylinder. The flow is found to have a transition from a steady axisymmetric state to a non-axisymmetric state exhibiting multiple azimuthal waves as the Mach number and the Reynolds number are varied. The flow field changes significantly with an increase in Ma for unsteady flow at higher Re. An increase in Ma increases the side wall azimuthal instability, as found in the perturbation contour plots and time series analysis. Further, we reconstruct phase portraits to show the dynamics of the flow finally becoming chaotic as the Reynolds number is increased to 5000. Finally, we support the argument with Lyapunov exponents for the higher Re samples. The Lyapunov Exponent is found to increase with Ma.

Transition to the Fluid Dynamic Limit: Mathematical Models and Simulation Results

Numerical simulations of standard situations in the transition region from gas kinetics to fluid dynamics at small Mach numbers indicate a clear dependence of the simulation results on the underlying kinetic model (here: nonlinear and linearized Boltzmann collision operator vs. BGK relaxation model). We develop an improved mathematical framework (trace theory) to explain these differences. In particular we reveal certain deficiencies for the classical BKG system as well as for the standard Navier Stokes approach.

One-dimensional mechanism of gaseous deflagration-to-detonation transition

A one-dimensional mechanism of deflagration to detonation transition is identified and investigated by an asymptotic analysis in the double limit of large activation energy and small Mach number of the laminar flame velocity. The unsteady analysis concerns the self-accelerating tip of an elongated flame in a smooth walled tube. The flame on the tip, considered as plane and orthogonal to the tube axis, is pushed from behind by the longitudinal flow resulting from the cumulative effect of the radial flows of burned gas issued from the lateral flame of the finger-like front (called backflow in the following). The analysis of the one-dimensional dynamics is performed by coupling the flame structure with the downstream-running compression waves propagating in the external flows. A critical elongation is identified from which the slightest increase in elongation leads to a pressure runaway producing the flame blow-off. The dynamics of the inner structure of the laminar flame on the tip which is accelerated by the self-induced backflow is characterized by a finite-time singularity of the reacting flow in the form of a dynamical saddle-node bifurcation.

Compressible two-dimensional turbulence: cascade reversal and sensitivity to imposed magnetic field

We study the impact of compressibility on two-dimensional turbulent flows, such as those modeling astrophysical disks. We demonstrate that the direction of cascade undergoes continuous transition as the Mach number Ma increases, from inverse at Ma = 0, to direct at . Thus, at comparable amounts of energy flow from the pumping scale to large and small scales, in accord with previous data. For supersonic turbulence with the cascade is direct, as in three dimensions, which results in multifractal density field. For that regime () we derive a Kolmogorov-type law for potential forcing and obtain an explicit expression for the third order correlation tensor of the velocity. We further show that all third order structure functions are zero up to first order in the inertial range scales, which is in sharp contrast with incompressible turbulence where the third order structure function, that describes the energy flux associated with the energy cascade is non-zero. The properties of compressible turbulence have significant implications on the amplification of magnetic fields in conducting fluids. We thus demonstrate that imposing external magnetic field on compressible flows of conducting fluids allows to manipulate the flow producing possibly large changes even at small Mach numbers. Thus Zeldovich’s antidynamo theorem, by which at Ma = 0 the magnetic field is zero in the steady state, must be used with caution. Real flows have finite Ma and, however small it is, for large enough values of I, the magnetic flux through the disk, the magnetic field changes the flow appreciably, or rearranges it completely. This renders the limit Ma → 0 singular for non-zero values of I. Of particular interest is the effect of the density multifractality, at which is relevant for astrophysical disks. We demonstrate that in that regime, in the presence of non-zero I the magnetic field energy is enhanced by a large factor as compared to its estimates based on the mean field. Finally, based on the insights described above, we propose a novel two-dimensional Burgers’ turbulence, whose three-dimensional counterpart is used for studies of the large-scale structure of the Universe, as a model for supersonic two-dimensional magnetohydrodynamic flows.

Tightly coupled hyperbolic treatment of buoyant two-phase flow and transport in porous media

Buoyant incompressible multiphase flow and transport in porous media typically is simulated by coupling an elliptic flow equation with a hyperbolic transport description. The tight coupling between flow and transport either requires a fully implicit solution algorithm or very small time steps, if solved sequentially. In any case, however, a large linear system has to be solved each time step.Here a new solution approach is devised, which relies on solving a coupled hyperbolic system of conservation laws with an explicit finite volume method. Consequently, only local operations have to be performed, which is a great advantage in case of massive parallel simulations and if GPUs are employed.The devised method is based on the isothermal Euler equations with momentum source terms accounting for resistance due to the porous medium and buoyancy. In this system the pressure is proportional to the density and in case of very small Mach numbers and relative density variation, the computed velocity field is divergence free and converges towards the mean total volumetric flow in the porous domain. To account for saturation transport, the system was augmented by an additional hyperbolic equation. In order to obtain the numerical fluxes, a characteristic based approximative Riemann solver was developed and 2nd order accuracy in space and time is achieved by piecewise linear reconstruction.Two-phase flow test cases with buoyant plumes and a lock exchange problem demonstrate that the devised hyperbolic approach recovers the correct incompressible solutions. Numerical experiments also demonstrate that the method is accurate and efficient.Thus the devised method is very promising, and conceptually it also can be applied for flow and transport in heterogeneous media; but it remains to be investigated how well it performs for such cases.

Study of air compressibility effects on the aerodynamic performance of the IEA-15 MW offshore wind turbine

With the gradual upscaling of offshore wind turbines, the blade size becomes large and the blade tip velocity reaches almost a Mach number of 0.3, the limit of the incompressible flow assumption.To simulate wind turbine flows, an incompressible solver was often a good choice as it gave good predictions and the Mach number of the wind turbine flows were small. When the Mach number is increased to be close to 0.3, the validity of incompressible flow may be questionable and thus needs to be checked. In this paper, an accurate wind turbine flow solution method is proposed to study the compressibility effects. First, the National Renewable Energy Laboratory (NREL) Phase VI wind turbine is used as an example to verify the method at a small Mach number. Through a comparative analysis between the compressible, the incompressible numerical results, and experimental data, the proposed method is found with a good simulation accuracy. Next, the flow fields of the large wind turbine IEA-15 MW, under different wind conditions are simulated. The local Mach number, density and pressure coefficient at different blade cross-sections of the compressible results are analyzed and compared to its incompressible ones, and the influences of compressibility on the thrust, torque and power of the wind turbine are quantified. The results show that at the rated wind speed, the thrust of the compressible flow simulation is 1.40 % higher than that of the incompressible one, and the torque of the compressible flow simulation is nearly 11.07 % higher than that of the incompressible one. As a result, the compressibility effects are recommended to be added in future studies of large wind turbines.

Constrained exact boundary controllability of a semilinear model for pipeline gas flow

AbstractWhile the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints.

Parabolic equation with arbitrary variations in the sound speed and Mach numbers of the medium velocity

Among computational techniques in atmospheric and ocean acoustics and other fields such as seismic wave propagation, the parabolic equation (PE) approach is one of most popular now. The PE is well suited to small computers, large domains, and high frequencies. It can handle many complicated phenomena such as atmospheric and ocean stratification and refraction, scattering by turbulence, internal waves, and other inhomogeneities, ground impedance and ocean bottom interactions, and propagation over slowly varying terrain and ocean bathymetry. PEs are usually formulated for small variations in the sound speed and small Mach numbers of the medium velocity. However, these assumptions might result in significant phase errors of propagating sound waves. In this presentation, a PE is formulated for arbitrary variations in the sound speed and arbitrary Mach numbers. This new PE is surprisingly simple and can be used in various fields of acoustics and physics. It can be implemented numerically with minimal modifications to existing Crank-Nicholson PE solvers described in Section 11.2 of Ostashev and Wilson, Acoustics in Moving Inhomogeneous Media, Second Edition (2015).

Equations of motion for weakly compressible point vortices.

Equations of motion for compressible point vortices in the plane are obtained in the limit of small Mach number, M, using a Rayleigh–Jansen expansion and the method of Matched Asymptotic Expansions. The solution in the region between vortices is matched to solutions around each vortex core. The motion of the vortices is modified over long time scales and . Examples are given for co-rotating and co-propagating vortex pairs. The former show a correction to the rotation rate and, in general, to the centre and radius of rotation, while the latter recover the known result that the steady propagation velocity is unchanged. For unsteady configurations, the vortex solution matches to a far field in which acoustic waves are radiated.This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.

The combined quasineutral and incompressible limit for the Navier–Stokes–Poisson system in critical spaces

In this paper, we study the quasineutral and incompressible limit for the Navier–Stokes–Poisson system with ill-prepared initial data in whole space. Based on Strichartz estimates for the Klein–Gordon equation and the pseudo-scaling property of modified Besov norms, we prove that the density tends to a constant, the compressible part of velocity vanishes, and the incompressible part of velocity converges to the solution of the incompressible Navier–Stokes equation. We get a relation of the lifespan of the solution between the Navier–Stokes–Poisson and the incompressible Navier–Stokes equations for small Debye length and low Mach number.

Inversion of the transverse force on a spinning sphere moving in a rarefied gas

The flow around a spinning sphere moving in a rarefied gas is considered in the following situation: (i) the translational velocity of the sphere is small (i.e. the Mach number is small); (ii) the Knudsen number, the ratio of the molecular mean free path to the sphere radius, is of the order of unity (the case with small Knudsen numbers is also discussed); and (iii) the ratio between the equatorial surface velocity and the translational velocity of the sphere is of the order of unity. The behaviour of the gas, particularly the transverse force acting on the sphere, is investigated through an asymptotic analysis of the Boltzmann equation for small Mach numbers. It is shown that the transverse force is expressed as $\boldsymbol{F}_L = {\rm \pi}\rho a^3 (\boldsymbol{\varOmega} \times \boldsymbol{v}) \bar{h}_L$ , where $\rho$ is the density of the surrounding gas, a is the radius of the sphere, $\boldsymbol {\varOmega }$ is its angular velocity, $\boldsymbol {v}$ is its velocity and $\bar {h}_L$ is a numerical factor that depends on the Knudsen number. Then, $\bar {h}_L$ is obtained numerically based on the Bhatnagar–Gross–Krook model of the Boltzmann equation for a wide range of Knudsen number. It is shown that $\bar {h}_L$ varies with the Knudsen number monotonically from 1 (the continuum limit) to $-\tfrac {2}{3}$ (the free molecular limit), vanishing at an intermediate Knudsen number. The present analysis is intended to clarify the transition of the transverse force, which is previously known to have different signs in the continuum and the free molecular limits.

Inviscid incompressible limit for compressible micro-polar fluids

In this paper we study the incompressible inviscid limit for a compressible micro-polar model. We prove that the weak solution of the compressible micro-polar system converges to the solution of the Navier–Stokes equations (Euler equations) in the limit of small Mach number (and vanishing viscosity).

Transient gas pipeline flow: analytical examples, numerical simulation and a comparison to the quasi-static approach

The operation of gas pipeline flow with high pressure and small Mach numbers allows to model the flow by a semilinear hyperbolic system of partial differential equations. In this paper we present a number of transient and stationary analytical solutions of this model. They are used to discuss and clarify why a PDE model is necessary to handle certain dynamic situations in the operation of gas transportation networks. We show that adequate numerical discretizations can capture the dynamical behavior sufficiently accurate. We also present examples that show that in certain cases an optimization approach that is based on multi-period optimization of steady states does not lead to approximations that converge to the optimal state.

Numerical study on the contribution of surface and volume components of flow-induced noise in baffle silencers

Baffle silencers are a well-known solution for noise mitigation in industrial applications. One of the issues concerning these devices is the flow-inducted noise produced when a non-laminar flow of the medium in the duct occurs. These situations occur, for example, in dedusting installations or exhaust systems with the high-speed flow (large Reynolds number of the turbulence and small Mach number). This kind of installation has a complex shape that causes a turbulent flow in the medium. Installing a baffle silencer in these conditions causes additional noise. This noise cannot be predicted by using a standard approach with equations for laminar flow conditions. This paper presents the first step of the research in this field. The first step is to find a relation between CFD simulations' results and self-noise of the baffle silencer. In this work, we use the formulation proposed by Proudman in 1952 to calculate the sound power generated by the flow. The formulation is based on the turbulent kinetic energy k and dissipation rate ε of the flow, which is calculated by CFD simulations. The resulting sound power level needs to be calibrated. The calibration method is developed and presented. The aim of this research is to design an experimental setup.

Numerical and experimental study on high-speed hydrogen–oxygen combustion gas flow and aerodynamic heating characteristics

The need to increase the payload capacity of the rockets motivates the development of high-power rocket engines. For a chemical propulsion system, this results in an increasing thermal load on the structure, especially the combustion chamber and nozzle must be able to withstand the extreme thermal load caused by high-temperature and high-pressure combustion gas. In order to protect the structure from the effect of increasing heat flux, it is necessary to counteract such effect with more advanced thermal management technology. This requires us to accurately predict the aerodynamic heating of the structure by high-temperature and high-speed combustion gas. In this study, a high-temperature combustion gas tunnel developed in the laboratory is used to produce high-speed combustion gas. Combined with the results of numerical calculation, the flow and aerodynamic heating characteristics of air and hydrogen–oxygen combustion gas under the same total temperature and pressure are analyzed and compared. The comparison revealed that the combustion gas flow in the nozzle has higher static temperature, velocity, and smaller Mach number. When the combustion gas flows around the sphere, the shock standoff distance and stagnation pressure are smaller than those of air, and the wall heat flux is much larger than that of air. The active chemical reaction in the combustion gas makes the aerodynamic heating of the structure more severe. Finally, through the analysis of a large amount of data, a semi-empirical formula for the heat flux of the stagnation point heated by a high-speed hydrogen and oxygen equivalent ratio combustion gas is obtained.

Amplification of Turbulence in Contracting Prestellar Cores in Primordial Minihalos

Abstract We investigate the amplification of turbulence through gravitational contraction of the primordial gas in minihalos. We perform numerical simulations to follow the cloud collapse, assuming polytropic equations of state for different initial turbulent Mach numbers and resolutions. We find that the turbulent velocity is amplified solely by gravitational contraction, and eventually becomes comparable to the sound speed, even for small initial turbulent Mach numbers (  0 ≳ 0.05 ). We derive an analytic formula for the amplification of turbulent velocity in a collapsing cloud, and find that our numerical results are consistent with the formula. These results suggest that the turbulence can play an important role in collapsing clouds for general cases.

Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics

In this paper, we consider the development of the two-dimensional discrete velocity Boltzmann model on a nine-velocity lattice. Compared to the conventional lattice Boltzmann approach for the present model, the collision rules for the interacting particles are formulated explicitly. The collisions are tailored in such a way that mass, momentum and energy are conserved and the H-theorem is fulfilled. By applying the Chapman–Enskog expansion, we show that the model recovers quasi-incompressible hydrodynamic equations for small Mach number limit and we derive the closed expression for the viscosity, depending on the collision cross-sections. In addition, the numerical implementation of the model with the on-lattice streaming and local collision step is proposed. As test problems, the shear wave decay and Taylor–Green vortex are considered, and a comparison of the numerical simulations with the analytical solutions is presented.

Effects of the Compressibility of Turbulence on the Dust Coagulation Process in Protoplanetary Disks

Abstract Planetesimals are believed to be formed by the coagulation of dust grains in the protoplanetary disk turbulence. However, the bouncing and fragmentation barriers have not been completely solved, particularly for silicate dust. To circumvent these barriers, the turbulent clustering of dust particles must be properly treated. According to the minimum-mass solar nebula (MMSN) model, the Mach number of the turbulence ranges from M rms ≃ 0.01–0.32, and thus the turbulence is often regarded as essentially incompressible. However, it has not been quantitatively investigated whether the incompressible limit is adequate for protoplanetary disk simulations. We therefore compare in this study the motions of inertial particles in direct numerical simulations (DNSs) of the Navier–Stokes equation between weakly compressible turbulence and incompressible turbulence. In the DNSs of compressible turbulence, we use an external force to set the total dissipation and the dilatational-to-solenoidal dissipation ratio. The DNSs reveal that despite the small Mach number M rms( ≲ 0.3), the compressible turbulence field notably differs from the incompressible field in terms of the density fluctuations, pressure fluctuations, and shocklet generation, depending on the ratio of the dilatational forcing. However, we quantitatively confirmed that these effects on the particle collision statistics are weak and that the motion of inertial particles in weakly compressible turbulence is dominated by the solenoidal velocity components. Therefore we can conclude that the incompressible assumption is appropriate for an investigation of the dust coagulation process in protoplanetary disk turbulence, as assumed in the MMSN model.