Low Mach Number Compressible Flow Research Articles

An implicit splitting scheme with characteristic boundary conditions for compressible reactive flows on unstructured grids

The computation of combustion instabilities requires efficient numerical solvers. For low Mach number compressible flows splitting schemes provide an adequate family of solution strategies. Here, the set of governing equations is separated into several subsystems, e.g. into an advective and acoustic subsystem. Advancements for such an algorithm are presented improving accuracy and stability. Moreover a new, gradient based approach is given for coupling the splitting scheme with characteristic boundary conditions on collocated unstructured grids. A large number of academic test cases is chosen to demonstrate the accuracy and performance of the proposed improved numerical scheme. In addition, the improved solver is tested in a complex, non-reactive LES of a swirl combustor on an unstructured grid.

Steady low Mach number flows: Identification of the spurious mode and filtering method

The aim of this article is to thoroughly identify the spurious mode that jeopardizes the convergence of usual upwind numerical schemes for compressible flows when the Mach number goes to 0. We show that this spurious mode is the long time limit of a wave system whose properties and discretization depend on the scheme used for the compressible system. Once this spurious mode is identified, a filtering method is developed for removing it from the solution of stationary low Mach number compressible flow. Numerical results confirm that at the price of the computation of a long time solution of the wave system, the accuracy of an inaccurate solution of a low Mach number compressible flow can be greatly improved by this filtering method.

Numerical simulation of flame acceleration and DDT(deflagration to detonation transition) in hydrogen-air mixtures with concentration gradients

ABSTRACT In this paper, a coupled new solver is used to simulate the acceleration and deflagration to detonation transition (DDT) of inhomogeneous mixtures flame under obstacle conditions. Analyzed the variations of flame front, pressure and density with time. The experimental results show that the vertical concentration gradient affects the chemical reaction rate in the combustion reaction zone, resulting in the flame front stretching along the “front up-back down.” At the same time, the flame propagation velocity is affected by obstacles. After multi-stage “acceleration-deceleration,” it reaches 2700 m/s at 1.23 m from the ignition center, and the pressure jumps to 5.0 MPa. Finally, the transition from detonation to detonation is completed. It is worth noting that the local explosion point is in the corner of the obstacle or pipe wall. This simulation method is compatible with low Mach number laminar flow, turbulent flow, and compressible reaction flow problems under high Mach number. It provides prospective prediction for flame acceleration and detonation formation during hydrogen combustion and detonation engine operation.

A fast transient solver for low-Mach number aerodynamics and aeroacoustics

The fractional step method is an efficient time advancement scheme for problems involving transient incompressible flow. The method is desirable over traditional SIMPLE-like algorithms for its non-iterative nature. This paper presents a technique that extends the fractional step method to low-Mach number compressible flow problems. The proposed pressure-velocity coupling and a transient SIMPLEC algorithm are validated and verified for a 1D acoustic wave propagation and a convective isentropic vortex to qualitatively evaluate their convergence properties. Next, both algorithms are used to predict aerodynamic and aeroacoustic behaviour of flow past a wall-mounted side mirror. The proposed scheme is stable at acoustic CFL number greater than one, and at convective Co number equal one. The L2 error norms of both schemes exhibit second order accuracy as grids are refined, while the new scheme requires significantly less computation effort for an equal error. The new fractional step method consumes only 20% to 35% of the computation effort that is required by the transient SIMPLEC algorithm for the same error, and is easily scalable to a greater number of CPU processors.

High-fidelity solver on polyhedral unstructured grids for low-Mach number compressible viscous flow

In this article, we developed an unstructured fluid solver based on finite volume framework for the low-Mach number compressible flows. The present method, so-called FVMS3 (Finite Volume method based on Merged Stencil with 3rd-order reconstruction) formulates two different numerical procedures for spatial reconstructions based on the quadratic polynomial which is performed by using least-square approximations on a merged stencil. In order to improve the reconstruction for discontinuities, we propose the limiting projection approach and smoothness adaptive fitting (SAF) scheme to suppress the numerical oscillation and limit the numerical dissipation. The resulting discretization algorithm that combines FVMS3 with SAF-based limiting projection scheme has third-order accuracy and resolves both smooth and non-smooth solutions with excellent quality. Additionally, a novel numerical model has been proposed by introducing the advection upstream splitting method (AUSM) flux into the pressure projection formulation which results in a unified scheme that works uniformly up to the incompressible limit. The fluid solver that integrates all above new efforts provides high-fidelity solutions for compressible viscous flows particularly for the low Mach regime. The performance of this new solver is verified by numerous benchmark tests. Our numerical results show that the proposed scheme gives accurate and robust solutions for a wide spectrum of test problems.

Coexistence of acoustic waves and turbulence in low Mach number compressible flows

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, modally equipartitioned compressible, and compressible wave. The distinction between these types of turbulence is investigated here applying different kinds of forcing. Scaling of density fluctuations with Mach number, comparison of the ratio of transverse velocity fluctuations to longitudinal fluctuations, and spectral decomposition of fluctuations are used to distinguish the nature of these solutions. From the study of the spatio-temporal spectra and correlation times, we quantify the contribution of the waves to the total energy of the system. Also, in the dynamics of a compressible flow, three associated correlation times are considered: the non-linear time of local interaction between scales, the sweeping time or non-local time of large scales on small scales, and the time associated with acoustic waves (sound). We observed that different correlation times dominate depending on the wave number (k), the Mach number, and the type of forcing.

Comparison of numerical methods for the solution of viscous incompressible and low Mach number compressible flow

The novelty of the paper is the application of the discontinuous Galerkin method (DGM) to the solution of viscous compressible flow with low Mach number (M≈10−2−10−3). The compressible Navier–Stokes equations are written in the conservative form using conservative variables. They are discretized by the DGM in space. In time we use either the backward difference formula (BDF) or the discontinuous Galerkin method. The resulting technique is called the space-time discontinuous Galerkin method (STDGM).The goal of the presented analysis is to prove the ability of the developed DGM to solve viscous compressible low Mach number flows described by the compressible Navier–Stokes equations without any modification and to obtain results comparable with simulations of incompressible viscous flow described by the incompressible Navier–Stokes equations, solved by the finite element method (FEM). We present numerical results for a flow in a simplified channel of the human vocal tract, for a flow around the airfoil NACA0012 and moreover for flow-induced airfoil vibrations. The results prove the robustness of the discontinuous Galerkin method with respect to the low Mach number.

Toward vortex identification based on local pressure-minimum criterion in compressible and variable density flows

We propose a dynamical vortex definition (the ‘$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$definition’) for flows dominated by density variation, such as compressible and multi-phase flows. Based on the search of the pressure minimum in a plane,$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$defines a vortex to be a connected region with two negative eigenvalues of the tensor$\unicode[STIX]{x1D64E}^{M}+\unicode[STIX]{x1D64E}^{\unicode[STIX]{x1D717}}$. Here,$\unicode[STIX]{x1D64E}^{M}$is the symmetric part of the tensor product of the momentum gradient tensor$\unicode[STIX]{x1D735}(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D66A})$and the velocity gradient tensor$\unicode[STIX]{x1D735}\unicode[STIX]{x1D66A}$, with$\unicode[STIX]{x1D64E}^{\unicode[STIX]{x1D717}}$denoting the symmetric part of momentum-dilatation gradient tensor$\unicode[STIX]{x1D735}(\unicode[STIX]{x1D717}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D66A})$, and$\unicode[STIX]{x1D717}\equiv \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D66A}$, the dilatation rate scalar. The$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$definition is examined and compared with the$\unicode[STIX]{x1D706}_{2}$definition using the analytical isentropic Euler vortex and several other flows obtained by direct numerical simulation (DNS) – e.g. liquid jet breakup in a gas, a compressible wake, a compressible turbulent channel and a hypersonic turbulent boundary layer. For low Mach number ($M\lesssim 5$) compressible flows, the$\unicode[STIX]{x1D706}_{2}$and$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$structures are nearly identical, so that the$\unicode[STIX]{x1D706}_{2}$method is still valid for low$M$compressible flows. But, the$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$definition is needed for studying vortex dynamics in highly compressible and strongly varying density flows.

Multiphase computational method for thermal interactions between compressible fluid and arbitrarily shaped solids

SummaryA numerical method was investigated for multiphase fields consisting of compressible gas and arbitrarily shaped solids. Since the proposed model is based on a one‐fluid model in which variables are averaged according to the phase fractions in the computational cells; it enables us to estimate gas‐solid momentum and thermal interactions without setting up adapting grids even if the solids have extremely complicated shapes. The governing equations are derived with the characteristics of an ideal gas assuming the specific heat to be uniform in the multiphase field. The derived equations in conservative form are discretized with a finite volume method. In addition, the pressure is calculated implicitly in a similar way to incompressible flow solvers. Because of these improvements, the proposed method allows us to calculate low Mach number compressible flows free from the Courant‐Friedrichs‐Lewy condition based on the speed of sound and to conserve the mass more accurately. To confirm the validity of the proposed method, it was applied to natural convection around an isothermal cylinder and a heat‐conducting pipe. In comparison with previous studies, it was confirmed that the gas flows and temperature distributions are predicted reasonably. In addition, a numerical experiment was conducted under more complicated conditions, namely, gas leaking from a container including heat sources. As a result, it was demonstrated that the proposed method enables us to predict unsteady variations of pressure and temperature distributions in the container due to the leakage while still conserving mass accurately.

Large eddy simulation of a transitional, thermal Blasius flow at low Reynolds number

Large eddy simulation (LES) of a transitional, thermal Blasius flowfield at a sufficiently low Reynolds number along an isothermal, flat plate is presented herein. Prior experiments observed that boundary layer transition is induced by the presence of streamwise vortex instability caused by the complex interaction between thermal buoyancy and forced convection dynamics. The maximum Grashof and Reynolds numbers employed in the LES were approximately 1.05×1011 and 1.18×105, respectively. To further enhance the accuracy, computational efficiency, and numerical stability, the LES solved the low-Mach number compressible flow governing equations, which included fluctuating density effects and pressure-density decoupling. For the subgrid scale (SGS) closure, a locally dynamic Smagorinsky SGS model was implemented into the LES solver to enable the backscatter phenomenon intrinsic to transitional boundary layer flows. The LES accurately predicted the onset of streamwise vortex instability and the eventual three-dimensional vortex breakdown of the underlying mean flow into a fully developed turbulent boundary layer, when compared to previously measured data. In the developed turbulence region, quadrant analyses indicated Q2 and Q4 events dominated the contribution to the Reynolds shear stress in the near-wall region, whereas Q1 and Q3 events contributed considerably to the wall-normal turbulent heat flux. And, as a result of the instability within the conduction layer, the layer erupts and intermittently releases buoyant thermal plumes. These recurring intermittent events inside the conduction layer deflect and deform the flowfield quantities near the wall, resulting in a plurality of peculiar peaks and shear layers inside the boundary layer. Furthermore, these buoyant thermal plumes are the dominant turbulence production mechanism farther away from the wall in the downstream region of the developed turbulent boundary layer flow. Near the wall, however, forced convection turbulent flow effects were observed, in which the shear production term was the primary contributor to the generation of turbulent kinetic energy, and quasi-streamwise and horseshoe-like vortex structures were observed in the developed turbulence region.

Wall-resolved large eddy simulation of turbulent mixed-convection heat transfer along a heated vertical flat plate

The present study aims to assess the predictive capabilities of wall-resolved large eddy simulation (LES) in computing the fluid flow and heat transfer characteristics in a mixed-convection turbulent boundary layer that developed along a large flat plate vertically mounted in air. The maximum Rayleigh number was approximately 3×1011, which resulted in fully developed turbulence conditions. To enhance the accuracy, computational efficiency, and numerical stability, the LES solved the low-Mach number compressible flow governing equations, which included fluctuating density effects and pressure-density decoupling. For the subgrid scale (SGS) closure, a locally dynamic Smagorinsky SGS model was implemented into the LES solver to enable the backscatter phenomenon intrinsic to transitioning boundary layer flows. The LES illustrated exceptional agreement with the statistics profiles in the boundary layer obtained with the experiments of previous studies, which showed sensitivity to freestream conditions. In particular, the LES correctly predicted the turbulent heat flux in the streamwise direction near the heated surface. Furthermore, the LES captured the rapid changes in spectra of fluctuating temperature and velocities due to the delay of transition with increasing freestream velocity.

Low Mach number algorithm for droplet-laden turbulent channel flow including phase transition

In this study we propose a new numerical algorithm for droplet-laden turbulent channel flow with phase transitions at low Mach numbers. The carrier gas is treated as compressible flow. In order to avoid very small time steps at low Mach numbers that would arise from stability requirements associated with explicit time-stepping we propose a new semi-explicit time integration method, applied to the low Mach number compressible flow equations. We perform a perturbation analysis in powers of the Mach number of the system of governing equations. The obtained decomposition of pressure into a space-independent part and a hydrodynamic part permits to apply a special pressure-based time integration algorithm for compressible flows at low Mach numbers. An important feature of the new numerical approach is the independence of the maximum allowed time step on the Mach number. In this study we validate the new method by comparing it with a fully explicit code for compressible flow at general Mach numbers showing a good agreement in all quantities of interest. The differences between the results of the two codes are on the order of the square of the Mach number caused by the disregard of high-order terms in the Mach number in the new algorithm. The relative difference found for a specific low value of the Mach number of 0.05 is on the order of 1% for instantaneous and mean quantities of the two phases. We also quantify the efficiency of the new algorithm by comparing the computational time it takes to simulate one time unit with both codes.

Lattice Boltzmann study on thermoacoustic onset in a Rijke tube

Nonlinear thermoacoustic self-excited onset was numerically studied in this work. A lattice Boltzmann model for viscous compressible flow and the implicit-explicit finite difference method were used to develop a solver. Nonlinear onset in an open-open Rijke tube with a constant-temperature stack was simulated with the solver. Based on the numerical results, overall onset process and self-excited standing wave in the Rijke tube are observed. The length of the Rijke tube along the x-direction covers a 1/4 wavelength of the standing wave and the main component of this standing wave is 171.2 Hz. These results agree well with the theoretical prediction. Instantaneous velocity and temperature fields at several phases under the limit cycle are presented and discussed. The maximal Mach number is about 0.035, indicating that the flow in the Rijke tube is a low Mach number compressible flow. This solver can also be applied for simulations of some other complex flows, such as the flow in porous media stack in thermoacoustic engine.

Identification of aero-acoustic scattering matrices from large eddy simulation. Application to a sudden area expansion of a duct

A methodology is presented which allows to determine the coefficients of transmission and reflection of plane acoustic waves at flow discontinuities in piping systems by combining large eddy simulation (LES) of turbulent compressible flows with system identification. The method works as follows. At first, an LES with external, broadband excitation of acoustic waves is carried out. Time series of acoustic data are extracted from the computed flow field and analyzed with system identification techniques in order to determine the acoustic scattering coefficients for a range of frequencies. The combination of broadband excitation with highly parallelized LES makes the overall approach quite efficient, despite the difficulties associated with simulation of low-Mach number compressible flows. The method is very general, here it is applied to study the scattering behavior of acoustic waves at a sudden change in cross-section in a duct system. The complex aero-acoustic interactions between acoustic waves and free shear layers are captured in detail by high resolution compressible LES, such that the scattering coefficients can be determined accurately from first principles. In order to demonstrate the reliability and accuracy of the method, the results for the scattering behavior and the acoustic impedance are presented and physically interpreted in combination with several analytical models and experimental data.

Fourth order compact finite volume scheme on nonuniform grids with multi-blocking

We have developed a fourth order compact finite volume method for the solution of low Mach number compressible flow equations on arbitrary nonuniform grids. The formulation presented here uses collocated grid that preserves fourth order accuracy on nonuniform meshes. This was achieved by introduction of a new fourth order method for calculation of cell and face averaged metrics. A special treatment of nonlinear terms is used to guarantee the stability of the fourth order compact method. Moreover an approach for applying this method to multi-block domains is presented for complicated geometries and parallel processing applications. Several test cases including the flow in a lid-driven cavity, laminar flow over a flat plate, decay of the Taylor vortex, and transient flow behind a backward facing step show the accuracy, efficiency, and stability of the method in single and multi-block domains. To the best of our knowledge, this is the first symmetric compact method in the finite volume form that can achieve full fourth order accuracy in solving fluid flow equations on nonuniform collocated grids without any stability problems.

Incompressible Criterion and Pressure Drop for Gaseous Slip Flow in Circular and Noncircular Microchannels

Slip flow in various noncircular microchannels has been further examined, and a simple model for a normalized Poiseuille number is proposed. As for slip flow, no solutions or graphical and tabulated data exist for most geometries; the developed simple model fills this void and can be used to predict the Poiseuille number, mass flow rate, tangential momentum accommodation coefficient, pressure distribution, and pressure drop of slip flow in noncircular microchannels by the research community for the practical engineering design of microchannels. The incompressible flow criterion for gas flow in microchannels is given. A Mach number less than 0.3 is not sufficient to ensure that the flow is incompressible. Compressibility depends on the product of two dimensionless parameters: L/L(DRe)(DRe) and Ma (Arkilic et al., 1997, “Gaseous Slip Flow in Long Microchannels,” J. Microelectromech. Syst., 6(2), pp. 167–178). Some flows where Ma < 0.3 are low speed compressible flows. A fresh general pressure drop model for isothermal low Mach number compressible flow in microchannels is proposed. If the pressure drop is less than 10% of the outlet pressure, the flow can be considered as incompressible for practical engineering applications. This paper improves and extends previous studies on slip flow in noncircular microchannels.

Heat transport mechanisms of low Mach number turbulent channel flow with spanwise wall oscillation

Large eddy simulation (LES) of low Mach number compressible turbulent channel flow with spanwise wall oscillation (SWO) is carried out. The flow field is analyzed with emphases laid on the heat transport as well as its relation with momentum transport. When turbulent coherent structures are suppressed by SWO, the turbulent transports are significantly changed, however the momentum and heat transports change in the same manner, which gives the evidence of inherently consistent transport mechanisms between momentum and heat in turbulent boundary layers. The Reynolds analogies of all the flow cases are quite good, which confirms again the fact that the transport mechanisms of momentum and heat are consistent, which gives theoretical support for controlling the wall heat flux control by using the drag reducing techniques.

Extension of the pressure correction method to zero-Mach number compressible flows

In the present paper, the classical pressure correction method was extended into low Mach number compressible flow regime by integrating equation of state into SIMPLE algorithm. The self-developed code based on this algorithm was applied to predicting the lid-driven cavity flow and shock tube problems, and the results showed good agreement with benchmark solutions and the Mach number can reach the magnitude of as low as 10−5. The attenuation of sound waves in viscous medium was then simulated. The results agree well with the analytical solutions given by theoretical acoustics. This demonstrated that the present method could also be implemented in acoustics field simulation, which is crucial for thermoacoustic simulation.

Asymptotic and numerical analysis of an inviscid bounded vortex flow at low Mach number

The inviscid low Mach number compressible flow developing from a plane incompressible vortex with constant density in a bounded domain is studied. A reference solution for this model flow is obtained by two-time scale asymptotic development in the zero Mach number limit. The solution can be decomposed into variations with a slow convective and a fast acoustic time scale. A selection of numerical schemes widely used to solve unsteady compressible low Mach number flows has been implemented to simulate this flow for two Mach number values, M = 0.1 and M = 0.01. The ability of the schemes to predict low Mach number flows and their behavior with the value of the Mach number have been analyzed.

Forms of Convection and Quadratic Conservative Finite Difference Schemes for Low-Mach Number Compressible Flow Simulations

After studying forms of convection in governing equations for compressible fluid flow, a new and useful skew-symmetric form was proposed. This leads to quadratic conservative schemes of convection for low-Mach number compressible fluid flow simulations. Then, commutable divergence, advection and skew-symmetric forms of convection were indicated for standard finite difference method in regular and staggered grid systems. Also, convection schemes for compact finite difference method in regular and staggered grid systems were indicated. In order to check the quadratic conservation property of the schemes, a numerical test was done on a three-dimensional periodic inviscid flow. The computational results have shown that the commutable convection schemes for standard finite difference method have quite good conservative property. For compact finite difference method, only the schemes with the proposed skew-symmetric form have acceptable conservative property.