Navier-Stokes Characteristic Boundary Conditions Research Articles

Navier-Stokes characteristic boundary conditions for high-enthalpy compressible flows in thermochemical non-equilibrium

This fundamental study presents Navier-Stokes characteristic boundary conditions (NSCBCs) for high-enthalpy hypersonic flows in thermochemical non-equilibrium. In particular, the relevant locally one-dimensional inviscid (LODI) relations are derived within a two-temperature framework for high-enthalpy hypersonic flows undergoing finite-rate thermochemical processes, including air dissociation and vibrational relaxation. Using these LODI relations, a set of NSCBCs are proposed and later demonstrated in canonical test cases, including the interaction of homogeneous isotropic turbulence with a shock wave subject to high-enthalpy thermochemical non-equilibrium effects.

Flame-acoustic interactions in high-pressure H2/air diffusion flames

Understanding flame-acoustics interaction (FAI) that can potentially trigger thermoacoustic instabilities is of critical importance to mitigate the serious side effects of pressure oscillations. While FAI has been studied extensively in premixed flames, comparatively less analysis has been conducted to explore FAI mechanisms in non-premixed flames, especially under high-pressure conditions. This study aims to numerically investigate FAI in high-pressure hydrogen/air counterflow diffusion flames, with special attention on the impact of pressure on FAI. To this end, a fully compressible unsteady counterflow solver combined with Navier-Stokes Characteristic Boundary Conditions (NSCBC) is employed to capture the reflection of the acoustic waves at the boundaries. The results show that for all ambient pressures ( 1 ≤ P a ≤ 60 bar) considered here, real-gas effects on flame structures are negligible and that increasing P a would lead to a power-law dependence of density gradient on P a with an exponent of 1.5. Furthermore, all tested cases show acoustic growth with time under fully reflecting boundary conditions, which is found to be pressure dependent. The flames become less resilient to FAI when the pressure (or density gradient) is increased. The results from the spectral analysis show that for all ambient pressures, there is only a single dominant frequency at the low-density fuel side and two dominant frequencies at the high-density oxidiser side. This indicates that the pressure oscillation is more sinusoidal at the H 2 side than that at the air side. Moreover, the phase space portraits of pressure signals indicate that the dynamics of the system are becoming periodic and approaching a limit cycle. It is found that the integrated heat release rate and the pressure fluctuations are partially in phase, which is responsible for the growth of high-frequency acoustic waves at both fuel and oxidiser sides. On the other hand, the significant density gradient contributes to the growth of low-frequency acoustic waves at the oxidiser side due to the high degree of acoustic reflectivity at the density boundary.

A polynomial-correction Navier-Stokes characteristic boundary condition

In an effort to assess the efficacy of different non-reflecting boundary conditions (NRBCs) in high-order solvers, several NRBCs have been implemented in the context of the discontinuous Galerkin (DG) discretization. The methods considered are based on buffer zone techniques and boundary-imposed conditions. The former mainly utilizes the sponge layer (SL), the perfectly matched layer (PML) and the supersonic layer (SSL), while the latter uses a Riemann extrapolation (RE) technique and a novel polynomial-correction Navier-Stokes characteristic boundary condition (PC-NSCBC). The developed PC-NSCBC method reconstructs an explicit boundary state, instead of modifying the equations on the boundary or specifying ad hoc compatibility conditions on corners and edges. This property pairs well with a Riemann solver, as is typical in a DG method. Moreover, the boundary state can be assembled either from a one-dimensional perspective (NSCBC1D) or using partial knowledge of multi-dimensional effects (NSCBC3D).Four different benchmark problems are considered: a 1D linear acoustic pulse in a quiescent flow, a 2D non-linear pressure pulse in a quiescent flow, an isentropic vortex and a shear-flow vortex shedding instability. Results imply that mainly the PML performs better than most other approaches. This is especially true for acoustically-dominated flows. Alternatively, for flow conditions predominantly less acoustic in nature and strongly non-linear, instabilities in the PML may arise. The SL and SSL both suggest that a larger layer is required to deliver comparable results to the PML. The proposed (PC-)NSCBC methods demonstrate reasonable success with no instabilities reported, which is impressive given the comparatively less computational effort required. Partial inclusion of the transverse terms in the NSCBC3D noticeably enhances the output of the NSCBC1D, albeit further investigation is required to determine an optimal, universal expression for its transverse tuning coefficient.

Development and validation of Navier–Stokes characteristic boundary conditions applied to turbomachinery simulations using the lattice Boltzmann method

AbstractThis article reports a procedure to implement as well as to validate non‐reflecting boundary conditions applied for turbomachinery simulations, using Navier–Stokes characteristic boundary conditions in a compressible lattice Boltzmann solver. The implementation of both an inlet condition imposing total pressure, total temperature, and flow angles, as well as an outlet condition imposing a static pressure profile that allows the simulation to reach a simplified radial equilibrium, is described within the context of a lattice Boltzmann approach. The treatment at the boundaries relies on the characteristic methodology to derive conditions which are non‐reflecting in terms of acoustics and is also compatible with turbulence injection at the inlet. These properties are evaluated on test cases of increasing complexity, ranging from a simple 2D periodic domain to an S‐duct stage with turbulence injection.

An approach for permeable boundary conditions in SPH

Smoothed Particle Hydrodynamics (SPH) has intrinsic advantages over the Finite Volume (FV) method for predicting multiphase flows, since the need for fluid phase interface reconstructions is avoided due to the natural advection of those interfaces based on the Lagrangian description of the control volumes. However, permeable boundary conditions are less straightforward to handle with SPH, even though domain inlets and outlets are needed in most engineering problems. In this work we propose a new framework for permeable boundary conditions for the Weakly Compressible SPH based on ghost particles. The major characteristics of our approach are the application of Navier Stokes Characteristic Boundary Conditions (NSCBC) and a particle mass generation and removal procedure at the boundary. This approach enables the application of non-reflective boundary conditions and redundantizes the need for advecting ghost particles from an external buffer zone to the internal domain. Additionally, the particle mass variation can exactly be balanced with the mass flux defined at the boundary. Furthermore, we are proposing a novel approach to set the ghost particle state based on an extrapolation of the state of particles located in the internal domain and on the domain boundary. We validate the proposed framework for permeable boundaries with one- and two-dimensional test cases. The major findings are as follows. Due to the particle mass generation and removal at the domain boundary, spurious oscillations are introduced due to a non-optimal particle distribution. Those oscillations are reduced in our work to an acceptable level (1×10−5 to 1×10−4 relative to the values of the fluid dynamics variables) using existing stabilization schemes as well as the aforementioned novel methods for setting ghost particle states and particle masses at the permeable boundaries. We observe a very good agreement between the solutions obtained with the presented approach and analytical as well as numerical solutions obtained with the FV method. The proposed approach for setting the ghost particle states is indispensable for achieving a perfect balance between particle mass variation and mass flux across the permeable interface as well as the steady state and the temporal accuracy of the solution. Additionally, the pressure reflectivity at the permeable boundary is found to be in a very good agreement with the theory of the NSCBC method. Furthermore, for a SPH-FV coupled simulation we could demonstrate an excellent agreement compared to the FV method as well as the capability of our approach to handle significant flow unsteadiness and gradients at the permeable coupling interfaces without the introduction of spurious flow behavior.

Numerical simulation of non-isothermal drainage flow of a viscous gas from a narrow channel

Drainage of a compressed viscous gas from a semi-sealed narrow conduit to a large container is a model problem of fundamental interest to fluid production from subsurface porous rocks. The low Mach number drainage flow exhibits behaviors contrary to predictions from conventional incompressible flow approximations. This work employs direct numerical simulation to study non-isothermal drainage flow from a semi-sealed narrow channel to a laterally bounded large container. The Navier–Stokes characteristic boundary conditions are implemented at the open boundary downstream in the computational domain to avoid spurious wave reflection. For a laterally bounded container, wave transport dominates the drainage process. It is shown that in the absence of an imposed temperature drop, thermal effect slows down the drainage process with adiabatic walls. With large enough temperature drop, thermal effect speeds up the drainage, and it also produces excess mass beyond that of the isothermal model. These thermal effects are consistent with the previous results for a laterally unbounded container in a linear theory.

An open-source density-based solver for two-phase [formula omitted] compressible flows: Verification and validation

A density-based solver based on the OpenFoam library coupled with a tabulated equation of state is developed. The Navier–Stokes characteristic boundary condition is considered to guarantee the accuracy and stability of the simulation. Firstly, two validation test cases are considered to illustrate the solver capability. Then, the flows in two converging-diverging nozzles are simulated under supercritical and subcritical conditions. A good agreement in terms of pressure distribution is obtained for all nozzles. The bulk viscosity and turbulence effects are then carefully assessed. The bulk viscosity does not affect significantly the predictions of the new solver, whereas it is of prime importance to well model the thermal and flow fields up to the wall. Finally, it is recommended to use an unsteady density-based solver for CO2 flows in converging-diverging nozzles associated with a Homogeneous Equilibrium Model and the k-ω SST closure to get the best overall predictions.

Revised Three-Dimensional Navier-Stokes Characteristic Boundary Conditions for Intense Reactive Turbulence

The three-dimensional Navier-Stokes characteristic boundary conditions (3D-NSCBC), although physically reasonable and popular in many applications, may encounter the instability problem in simulating complex flows, especially for large Reynolds number reactive turbulence where locally the strong reversed flow appears at the outflow boundary surfaces. In the present work, a revised 3D-NSCBC strategy is proposed based on the kinematic relation in different moving coordinate systems. Following this strategy, a systematic formulation is presented for the outflow surface with local reversed flow and can be easily extended to the coupled edge and corner boundaries. Direct numerical simulation (DNS) tests of flow with different turbulence intensities are carried out. Compared with the conventional 3D-NSCBC, the newly proposed method exhibits satisfactory performance to confine numerical instability in the strong reversed flow region. The results confirm the robustness and effectiveness of this newly proposed algorithm.

Navier–Stokes Characteristic Boundary Conditions Using Ghost Cells

Solution methods for the compressible Navier–Stokes equations based on finite volume discretizations often implement boundary conditions using ghost cells outside of the computational domain. Filling the ghost cells using straightforward zeroth- or first-order extrapolation, although computationally expedient, is well known to fail even for some simple flows, especially when turbulent structures interact with the boundaries or if time-varying inflow conditions are imposed. The Navier–Stokes characteristic boundary condition approach provides more accurate boundary conditions, but requires the use of special discretizations at boundaries. The present paper develops a new technique based on the Navier–Stokes characteristic boundary condition approach to derive values for ghost cells that significantly improve the treatment of boundaries over simple extrapolation, but retain the ghost cell approach. It is demonstrated in the context of a Godunov integration procedure that the new method provides accurate results, while allowing the use of the same stencil and numerical methodology near the boundaries as in the interior.

On the robust, flexible and consistent implementation of time domain impedance boundary conditions for compressible flow simulations

The accurate simulation of compressible flows requires the appropriate modeling of the reflection of acoustic waves at the boundaries. In the present study we discuss time domain impedance boundary conditions (TDIBC). The formulation proposed allows to impose a desired reflection coefficient at the inflow and outflow boundaries. Our formulation is an extension of the well known Navier-Stokes characteristic boundary conditions. The frequency dependent reflections at the boundaries are implemented with a state-space model in the time domain. We provide a comprehensive discussion on how such state-space models can be constructed and interpreted. This discussion shows that the state-space description allows a robust and flexible implementation. It allows to consider complex reflection coefficients and account for non-constant CFD time steps in a straight forward manner. Furthermore, we prove analytically and demonstrate numerically that the formulation proposed is consistent, i.e. the formulation ensures that the flow simulation exhibits the reflection coefficient imposed accurately, as long as the waves impinging on the boundary are plane, and it prohibits drift of the mean flow variables. Finally, the boundary conditions are tested successfully for laminar and turbulent flows.

Improved Navier–Stokes Boundary Conditions Based on Generalized Characteristics

In this paper, the three-dimensional Navier–Stokes characteristic boundary conditions for large-eddy and aeroacoustic simulations are extended to curvilinear coordinates formulations. A robust way of treating the transverse and gradient terms on boundary planes is presented which is different from previous generalized characteristic boundary conditions. The performance of the new formulation is examined via four test problems: an inviscid convective vortex, a two-dimensional mixing layer, a Mach 0.75 round jet, and a Mach 0.51 nozzle/jet. For each test problem, the numerical schemes used to implement the boundary conditions, the numerical parameters employed, and the predicted three-dimensional flow fields are presented. Based on the numerical experiments conducted, the new boundary conditions show promise for high-fidelity simulations of compressible viscous flows.

The Realization of Non-reflecting Boundaries for Compressible Rayleigh-Taylor Flows with Variable Acceleration Histories

The Navier-Stokes characteristic boundary conditions (NSCBC) approach is extended to compressible Rayleigh-Taylor flow (CRT-F) with variable acceleration histories, whose time- and space- dependent open boundaries challenge the available methods. The non-reflecting boundary conditions are realized by combining CRTF's physical boundary conditions and NSCBC's idea, and by appending a dissipation region, where physics-consistent viscous terms are introduced to realize non-reflection without additional effect. Numerical tests confirm the effectiveness and robustness of newly proposed schemes.

LES of Gas Exchange in IC Engines

As engine technologies become increasingly complex and engines are driven to new operating points, understanding transient phenomena is important to ensure reliable engine operation. Unlike Reynolds Averaged Navier-Stokes (RANS) studies that only provide cycle-averaged information, Large Eddy Simulation (LES) studies are capable of simulating cycle-to-cycle dynamics. In this work, a finite difference based structured methodology for LES of IC engines is presented. This structured approach allows for an efficient mesh generation process and provides potential for higher order numerical accuracy. An efficient parallel scalable block decomposition is done to overcome the challenges associated with the low ratio of fluid elements to overall mesh elements. The motion of the valves and piston is handled using a dynamic cell blanking approach and the Arbitrary Lagrangian Eulerian (ALE) method, respectively. Modified three-dimensional Navier-Stokes Characteristic Boundary Conditions (NSCBC) are used in the simulation to prescribe conditions in the manifolds. The accuracy of the simulation framework is validated using various canonical configurations. Flow bench simulations of an axisymmetric configuration and an actual engine geometry are done with the LES methodology. Simulations of the gas exchange in an engine under motored conditions are also performed. Overall, good agreement is obtained with experiments for all the cases. Therefore, this framework can be used for LES of engine simulations. In the future, reactive LES simulations will be performed using this framework.

An implicit algorithm based on iterative modified approximate factorization method coupling with characteristic boundary conditions for solving subsonic viscous flows

An algorithm composed of an iterative modified approximate factorization (MAF(k)) method with Navier-Stokes characteristic boundary conditions (NSCBC) is proposed for solving subsonic viscous flows. A transformation on the matrix equation in MAF(k) is made in order to impose the implicit boundary conditions properly. To be in consistent with the implicit solver for the interior domain, an implicit scheme for NSCBC is formulated. The performance of the developed algorithm is investigated using spatially evolving zero pressure gradient boundary layer over a flat plate and a wall jet mixing with a cross flow over a flat plate with a square hole as the test cases. The numerical results are compared to the existing experimental datasets and a number of general correlations, together with other available numerical solutions, which demonstrate that the developed algorithm possesses promising capacity for simulating the subsonic viscous flows with large CFL number.

1208 THE EFFECT OF UPSTREAM DISTURBANCE ON THE ANGLE OF SOUND EMISSION IN A SUPERSONIC ROUND JET

The three-dimensional time-dependent compressible Navier-Stokes equations are numerically solved to study acoustic emission mechanisms in a supersonic round jet at high convective Mach numbers. A 5th-order compact upwind algorithm developed by Deng and Maekawa (1996)[1] is used for spatial derivatives and a 4th-order Runge-Kutta scheme for time advancement. The Navier-Stokes characteristic boundary conditions are used in the streamwise and radial directions and periodic boundary conditions in the azimuthal direction. Numerical results for the convective Mach number Mc = 0.97 are presented (Mc is defined by eq.(13) in Section2). Two different cases were investigated. The first case is the jet forced by the linear unstable modes. The second case is the jet flow forced randomly. The numerical results provide new physical insights into Mach wave generation nature in a supersonic round jet. Upstream disturbance conditions play an important role for the emission of the Mach waves in a supersonic jet. The numerical results show that the jet forced with a pair of first helical modes can indicate the elimination of Mach waves at restricted emission angles due to the interference of these modes. The pressure fluctuations generated by the growth of the opposite helical mode are lineally superposed into the jet near sound filed. Therefore, this result suggests that the direction of Mach wave radiation can be controlled by introducing of optimal helical modes in turbulent jets.

Three-dimensional boundary conditions for numerical simulations of reactive compressible flows with complex thermochemistry

The Navier–Stokes characteristic boundary conditions (NSCBC) is a very efficient numerical strategy to treat boundary conditions in fully compressible solvers. The present work is an extension of the 3D-NSCBC method proposed by Yoo et al. and Lodato et al. in order to account for multi-component reactive flows with detailed chemistry and complex transport. A new approach is proposed for the outflow boundary conditions which enables clean exit of non-normal flows, and the specific treatment of all kinds of edges and corners is carefully addressed. The proposed methodology is successfully validated on various challenging multi-component reactive flow configurations.

Large eddy simulation of a pulsed jet in cross-flow

AbstractThis study quantifies the mixing that results from a pulsed jet in cross-flow in the near jet region. By large eddy simulation computations, it also helps to understand the physical phenomena involved in the formation of the pulsed jet in cross-flow. The boundary conditions of the jet inlet are implemented via a Navier–Stokes characteristic boundary condition coupled with a Fourier series development. The signals used to pulse the jet inlet are a square or a sine wave. A new way of characterizing the mixing is introduced with the goal of easily interpreting and quantifying the complicated mixing process involved in a pulsed jet in cross-flow flow fields. Different flow configurations, pulsed and non-pulsed, are computed and compared, keeping the root mean square value of the signal constant. This comparison not only allows the characterization of the mixing but also illustrates some of the properties of the mixing characterization.

Flow streamline based Navier–Stokes Characteristic Boundary Conditions: Modeling for transverse and corner outflows

Some limitations of three dimensional Navier–Stokes Characteristic Boundary Conditions (3D-NSCBC) are discussed for flows traveling in a direction that is oblique to the boundary. To limit errors generated at boundaries with flows having any arbitrary direction, it is proposed to organize the wave decomposition in a coordinate system that is attached to the local flow streamline crossing the boundary, because some modeled expressions are not frame independent. Compared to previous 3D-NSCBC, the modified strategy accounting for oblique waves is found to improve the outflow treatment for transverse outgoing vortices, up to vortices crossing an outflow corner. The method is also applied to an expanding laminar flame.

Comparison of outflow boundary conditions for subsonic aeroacoustic simulations

SUMMARYAeroacoustics simulations require much more precise boundary conditions than classical aerodynamics. Two classes of non‐reflecting boundary conditions for aeroacoustics are compared in the present work: the characteristic analysis‐based methods and the Tam and Dong approach. In the characteristic methods, waves are identified and manipulated at the boundaries, whereas the Tam and Dong approach use modified linearized Euler equations in a buffer zone near outlets to mimic a non‐reflecting boundary. The principles of both approaches are recalled, and recent characteristic methods incorporating the treatment of transverse terms are discussed. Three characteristic techniques—the original Navier–Stokes characteristic boundary conditions (NSCBC) of Poinsot and Lele and two versions of the modified method of Yoo and Im—are compared with the Tam and Dong method for four typical aeroacoustics problems: vortex convection on a uniform flow, vortex convection on a shear flow, acoustic propagation from a monopole, and acoustic propagation from a dipole. Results demonstrate that the Tam and Dong method generally provides the best results and is a serious alternative solution to characteristic methods even though its implementation might require more care than the usual NSCBC approaches. Copyright © 2011 John Wiley & Sons, Ltd.

Erratum to: Geometrical Properties and Turbulent Flame Speed Measurements in Stationary Premixed V-flames Using Direct Numerical Simulation

Three dimensional, fully compressible direct numerical simulations (DNS) of premixed turbulent flames are carried out in a V-flame configuration. The governing equations and the numerical implementation are described in detail, including modifications made to the Navier–Stokes Characteristic Boundary Conditions (NSCBC) to accommodate the steep transverse velocity and composition gradients generated when the flame crosses the boundary. Three cases, at turbulence intensities, u′/sL, of 1, 2, and 6 are considered. The influence of the flame holder on downstream flame properties is assessed through the distributions of the surface-conditioned displacement speed, curvature and tangential strain rates, and compared to data from similarly processed planar flames. The distributions are found to be indistinguishable from planar flames for distances greater than about 17δth downstream of the flame holder, where δth is the laminar flame thermal thickness. Favre mean fields are constructed, and the growth of the mean flame brush is found to be well described by simple Taylor type diffusion. The turbulent flame speed, sT is evaluated from an expression describing the propagation speed of an isosurface of the mean reaction progress variable \(\tilde{c}\) in terms of the imbalance between the mean reactive, diffusive, and turbulent fluxes within the flame brush. The results are compared to the consumption speed, sC, calculated from the integral of the mean reaction rate, and to the predictions of a recently developed flame speed model (Kolla et al., Combust Sci Technol 181(3):518–535, 2009). The model predictions are improved in all cases by including the effects of mean molecular diffusion, and the overall agreement is good for the higher turbulence intensity cases once the tangential convective flux of \(\tilde{c}\) is taken into account.