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Navier-Stokes characteristic boundary conditions for high-enthalpy compressible flows in thermochemical non-equilibrium
... While initially proposed in pure finite-difference contexts, the NSCBC approach 40 has also been implemented in conjunction with the lattice Boltzmann method [23] and using ghost cells [24,25]. Very recently, Williams [26] proposed LODI-based NSCBC for hypersonic flows in thermochemical non-equilibrium. ...
... (27), and/or Eq. (28) is calculated using Eq. (26), and these equations are integrated to update the required variables (i.e. those that are not assigned by physical boundary conditions) at the boundary points. ...
... Energy exchange between these non-equilibrium internal degrees of freedom and the rotational-translational modes proves crucial not only for predicting heat fluxes but also for the evaluation of chemical production rates, with the effective vibrational excitation modulating the reaction rates through vibration-dissociation coupling. 59 For finite-rate thermochemical relaxation that proceeds on the advective timescale, namely Da ≃ 1, direct numerical solution of the relevant non-linear conservation laws generally proves necessary for characterizing the non-equilibrium flow physics [60][61][62] . ...
Hypersonic flight involves a variety of complex flow phenomena that directly impact the aerothermodynamic loading of high-speed vehicles. The turbulence encountered during a typical flight trajectory influences and interacts with the shock waves on and around the surface of a vehicle and its propulsion system, affecting both aerodynamic and power plant performance. These interactions can be studied by isolating a turbulent flow convected through a normal shock, commonly referred to as the canonical shock-turbulence interaction (STI) problem. Scale-resolving computational fluid dynamics (CFD) and linear interaction analysis (LIA) have been crucial in studying this problem and formulating scaling laws that explain the observed behavior. In this work, an extensive review of the theoretical (LIA) and numerical (CFD) work on the canonical STI is presented. The majority of the work conducted to date has focused on calorically perfect gases with constant heat capacities. However, in hypersonic flows, chemical and thermal non-equilibrium effects may alter the nature of the interaction. As a result, relevant LIA and CFD studies addressing high-enthalpy phenomena are also succinctly discussed.
A hypersonic, spatially-evolving turbulent boundary layer at Mach 12.48 with a cooled wall is analyzed by means of direct numerical simulations. At the selected conditions, massive kinetic-to-internal energy conversion triggers thermal and chemical nonequilibrium phenomena. Air is assumed to behave as a five species reacting mixture, and a two-temperature model is adopted to account for vibrational nonequilibrium. Wall cooling partly counteracts the effects of friction heating, and the temperature rise in the boundary layer excites vibrational energy modes while inducing mild chemical dissociation of oxygen. Vibrational non-equilibrium is mostly driven by molecular nitrogen, characterized by slower relaxation rates than the other molecules in the mixture. The results reveal that thermal nonequilibrium is sustained by turbulent mixing: sweep and ejection events efficiently redistribute the gas, contributing to the generation of a vibrationally under-excited state close to the wall, and an over-excited state in the outer region of the boundary layer.The tight coupling between turbulence and thermal effects is quantified by defining an interaction indicator. A modeling strategy for the vibrational energy turbulent flux is proposed, based on the definition of a vibrational turbulent Prandtl number. The validity of the Strong Reynolds Analogy under thermal nonequilibrium is also evaluated. Strong compressibility effects promote the translational-vibrational energy exchange, but no preferential correlation was detected between expansions/compressions and vibrational over-/under-excitation, as opposed to what has been observed for unconfined turbulent configurations.
The interaction between a weakly turbulent free stream and a hypersonic shock wave is investigated theoretically by using linear interaction analysis (LIA). The formulation is developed in the limit in which the thickness of the thermochemical nonequilibrium region downstream of the shock, where relaxation toward vibrational and chemical equilibrium occurs, is assumed to be much smaller than the characteristic size of the shock wrinkles caused by turbulence. Modified Rankine–Hugoniot jump conditions that account for dissociation and vibrational excitation are derived and employed in a Fourier analysis of a shock interacting with three-dimensional isotropic vortical disturbances. This provides the modal structure of the post-shock gas arising from the interaction, along with integral formulas for the amplification of enstrophy, concentration variance, turbulent kinetic energy (TKE), and turbulence intensity across the shock. In addition to confirming known endothermic effects of dissociation and vibrational excitation in decreasing the mean post-shock temperature and velocity, these LIA results indicate that the enstrophy, anisotropy, intensity, and TKE of the fluctuations are much more amplified through the shock than in the thermochemically frozen case. In addition, the turbulent Reynolds number is amplified across the shock at hypersonic Mach numbers in the presence of dissociation and vibrational excitation, as opposed to the attenuation observed in the thermochemically frozen case. These results suggest that turbulence may persist and get augmented across hypersonic shock waves despite the high post-shock temperatures.
SU2-NEMO, a recent extension of the open-source SU2 multiphysics suite’s set of physical models and code architecture, is presented with the aim of introducing its enhanced capabilities in addressing high-enthalpy and high-Mach number flows. This paper discusses the thermal nonequilibrium and finite-rate chemistry models adopted, including a link to the Mutation++ physio-chemical library. Further, the paper discusses how the software architecture has been designed to ensure modularity, incorporating the ability to introduce additional models in an efficient manner. A review of the numerical formulation and the discretization schemes utilized for the convective fluxes is also presented. Several test cases in two- and three-dimensions are examined for validation purposes and to illustrate the performance of the solver in addressing complex nonequilibrium flows.
A Mach-10 hypersonic boundary layer of air overriding a cold, isothermal, non-catalytic flat wall, and with a stagnation enthalpy of 21.6 MJ kg −1 , is analysed using direct numerical simulations. The calculations include multicomponent transport, equilibrium vibrational excitation and chemical kinetics for air dissociation. The initially laminar boundary layer undergoes transition to turbulence by the resonance of a two-dimensional mode injected by a suction-and-blowing boundary condition imposed over a narrow spanwise porous strip. The ensuing turbulent boundary layer has a momentum Reynolds number of 3826 near the outflow of the computational domain. The relatively low temperature of the free stream renders the air chemically frozen there. However, the high temperatures generated within the boundary layer by viscous aerodynamic heating, peaking at a wall-normal distance y 10-20 in semi-local viscous units, lead to air dissociation in under-equilibrium amounts equivalent to 4 %-7 % on a molar basis of atomic oxygen, along with smaller concentrations of nitric oxide, which is mainly produced by the Zel'dovich mechanism, and of atomic nitrogen, the latter being mostly in steady state. A statistical analysis of the results is provided, including the streamwise evolution of (a) the skin friction coefficient and dimensionless wall heat flux; (b) the mean profiles of temperature, velocity, density, molar fractions, chemical production rates and chemical heat-release rate; (c) the Reynolds stresses and root-mean-squares of the fluctuations of temperature, density, pressure, molar fractions and chemical heat-release rate; and (d) the temperature/velocity and mass-fraction/velocity correlations.
The interaction between an incident shock wave and a Mach-6 undisturbed hypersonic laminar boundary layer over a cold wall is addressed using direct numerical simulations (DNS) and wall-modeled large-eddy simulations (WMLES) at different angles of incidence. At sufficiently high shock-incidence angles, the boundary layer transitions to turbulence via breakdown of near-wall streaks shortly downstream of the shock impingement, without the need of any inflow free-stream disturbances. The transition causes a localized significant increase in the Stanton number and skin-friction coefficient, with high incidence angles augmenting the peak thermomechanical loads in an approximately linear way. Statistical analyses of the boundary layer downstream of the interaction for each case are provided that quantify streamwise spatial variations of the Reynolds analogy factors and indicate a breakdown of the Morkovin's hypothesis near the wall, where velocity and temperature become correlated. A modified strong Reynolds analogy with a fixed turbulent Prandtl number is observed to perform best. Conventional transformations fail at collapsing the mean velocity profiles on the incompressible log law. The WMLES prompts transition and peak heating, delays separation, and advances reattachment, thereby shortening the separation bubble. When the shock leads to transition, WMLES provides predictions of DNS peak thermomechanical loads within at a computational cost lower than DNS by two orders of magnitude. Downstream of the interaction, in the turbulent boundary layer, WMLES agrees well with DNS results for the Reynolds analogy factor, the mean profiles of velocity and temperature, including the temperature peak, and the temperature/velocity correlation.
Although classical WENO schemes have achieved great success and are widely accepted, they exhibit several shortcomings. They are too dissipative for direct simulations of turbulence and lack robustness when very-high-order versions are applied to complex flows. In this paper, we propose a family of high-order targeted ENO schemes which are applicable for compressible-fluid simulations involving a wide range of flow scales. In order to increase the numerical robustness as compared to very-high-order classical WENO schemes, the reconstruction dynamically assembles a set of low-order candidate stencils with incrementally increasing width. While discontinuities and small-scale fluctuations are efficiently separated, the numerical dissipation is significantly diminished by an ENO-like stencil selection, which either applies a candidate stencil with its original linear weight, or removes its contribution when it is crossed by a discontinuity. The background linear scheme is optimized under the constraint of preserving an approximate dispersion-dissipation relation. By means of quasi-linear analyses and practical numerical experiments, a set of case-independent parameters is determined. The general formulation of arbitrarily high-order schemes is presented in a straightforward way. A variety of benchmark-test problems, including broadband waves, strong shock and contact discontinuities are studied. Compared to well-established classical WENO schemes, the present schemes exhibit significantly improved robustness, low numerical dissipation and sharp discontinuity capturing. They are particularly suitable for DNS and LES of shock-turbulence interactions.
A set of direct numerical simulations of isotropic turbulence passing through a nominally normal shock wave is presented. Upstream of the shock, the microscale Reynolds number is 40, the mean Mach number is 1.3–6.0, and the turbulence Mach number is 0.16–0.38. It is shown that the Kolmogorov scale decreases during the shock interaction, which implies that the grid resolution needed to resolve the viscous dissipation is finer than that used in previous studies. This leads to some qualitative differences with previous work, e.g., a rapid increase in the streamwise vorticity variance behind the shock and large anisotropy of the postshock Reynolds stresses. The instantaneous structure of the shock/turbulence interaction is examined using averages conditioned on the instantaneous shock strength. For locally strong compressions, the flow is characterized by overcompression, followed by an expansion. At points where the shock is locally weak, the profiles differ qualitatively depending on the strength of the incoming turbulence relative to the strength of the shock, as measured by the turbulence and mean Mach numbers, respectively. In the wrinkled shock regime, these profiles are discontinuous and the shock has a simple topology. In the broken shock regime, the weak interaction profiles are smooth without any discontinuity.
Chemical-kinetic parameters governing the Row in the shock layer over a heat shield of a blunt body entering Earth's atmosphere from a hyperbolic orbit are derived. By the use of the assumption that the heat shield is made of carbon phenolic and by allowing for an arbitrary rate of pyrolysis-gas injection, chemical reactions occurring in the shock layer are postulated, and the collision integrals governing the transport properties, the rate coefficients of the reactions, and the parameters needed for the bifurcation model and for the finite-rate kinetic wall boundary conditions are determined using the best available techniques. Sample flowfield calculations are performed using this set of parameters to show that the heating and surface removal rates are substantially smaller than calculated using the existing set of such parameters and traditional assumptions of gas-surface equilibrium and quasi-steady-state ablation.
Direct numerical simulation and inviscid linear analysis are used
to
study the interaction of a normal shock wave with an isotropic turbulent
field
of vorticity and entropy fluctuations. The role of the upstream entropy
fluctuations is emphasized. The upstream correlation between the vorticity
and
entropy fluctuations is shown to strongly influence the evolution of the
turbulence across the shock. Negative upstream correlation between
u′ and T′ is seen to enhance the amplification
of the
turbulence kinetic energy, vorticity and thermodynamic fluctuations across
the shock
wave. Positive upstream correlation has a suppressing effect. An explanation
based
on the relative effects of bulk compression and baroclinic torque is proposed,
and a
scaling law is derived for the evolution of vorticity fluctuations across
the shock.
The validity of Morkovin's hypothesis across a shock wave is examined.
Linear
analysis is used to suggest that shock-front oscillation would invalidate
the
relation between urms and
Trms, as expressed by the hypothesis.
Compressible flows with r.m.s. velocities of the order of the speed of sound are studied with direct numerical simulations using a pseudospectral method. We concentrate on turbulent homogeneous flows in the two-dimensional case. The fluid obeys the Navier-Stokes equations for a perfect gas, and viscous terms are included explicitly. No modelling of small scales is used. We show that the behaviour of the flow differs sharply at low compared with high r.m.s. Mach number Ma, with a transition at Ma = 0.3. In the large scales, temporal exchanges between longitudinal and solenoidal modes of energy retain an acoustical character; they lead to a slowing down of the decrease of the Mach number with time, which occurs with interspersed plateaux corresponding to quiescent periods. When the flow is initially supersonic, the small scales are dominated by shocks behind which vortices form. This vortex production is particularly prominent when two strong shocks collide, with the onset of shear turbulence in the region downstream of the collision. However, at the resolutions reached by our code on a 256 × 256 uniform grid, this mechanism proves insufficient to bring vortices into equipartition with shocks in the small-scale tail of the energy spectrum.
Interaction of isotropic quasi-incompressible turbulence with a weak shock wave was studied by direct numerical simulations. The effects of the fluctuation Mach number M t of the upstream turbulence and the shock strength M ² 1 — 1 on the turbulence statistics were investigated. The ranges investigated were 0.0567 ≤ M t ≤ 0.110 and 1.05 ≤ M 1 ≤ 1.20. A linear analysis of the interaction of isotropic turbulence with a normal shock wave was adopted for comparisons with the simulations.
Both numerical simulations and the linear analysis of the interaction show that turbulence is enhanced during the interaction with a shock wave. Turbulent kinetic energy and transverse vorticity components are amplified, and turbulent lengthscales are decreased. The predictions of the linear analysis compare favourably with simulation results for flows with M ² t < a ( M ² 1 — 1) with a ≈ 0.1, which suggests that the amplification mechanism is primarily linear. Simulations also showed a rapid evolution of turbulent kinetic energy just downstream of the shock, a behaviour not reproduced by the linear analysis. Investigation of the budget of the turbulent kinetic energy transport equation shows that this behaviour can be attributed to the pressure transport term.
Shock waves were found to be distorted by the upstream turbulence, but still had a well-defined shock front for M ² t < a ( M ² 1 — 1) with a ≈ 0.1). In this regime, the statistics of shock front distortions compare favourably with the linear analysis predictions. For flows with M ² t > a ( M ² 1 — 1 with a ≈ 0.1, shock waves no longer had well-defined fronts: shock wave thickness and strength varied widely along the transverse directions. Multiple compression peaks were found along the mean streamlines at locations where the local shock thickness had increased significantly.
We present an updated version of the open-source Hypersonics Task-based Research (HTR) solver for hypersonic aerothermodynamics. The solver, whose first version was presented in Di Renzo et al. (Comput. Phys. Commun. 255, 2020), is designed for direct numerical simulation (DNS) of canonical hypersonic flows at high Reynolds numbers in which thermo-chemical effects induced by high temperatures are relevant. The solver relies on high-order spatial discretization on structured meshes and efficient time integrators for stiff systems within the Regent/Legion software stack, which makes the code highly portable and scalable in CPU and GPU-based supercomputers. The new version herein presented includes several optimizations and new tools for data analysis, along with novel user option for hybrid skew-symmetric/targeted essentially non-oscillatory numerics, to offer higher computational efficiency and lower numerical dissipation at moderate Mach numbers, inclusion of a new combustion mechanism for methane and oxygen, and new recycling-rescaling inlet boundary conditions targeted to the simulation of fully developed turbulent boundary layers.
Program summary
Program Title: Hypersonics Task-based Research solver
CPC Library link to program files:
Developer’s respository link: https://github.com/stanfordhpccenter/HTR-solver.git
Licensing provisions: BSD 2-clause
Programming language: Regent, C++, and CUDA
Journal Reference of previous version: Di Renzo, M., Fu, L., & Urzay, J. (2020). HTR solver: An open-source exascale-oriented task-based multi-GPU high-order code for hypersonic aerothermodynamics. Computer Physics Communications, 107262.
Does the new version supersede the previous version?: Yes
Reasons for the new version: Release of new features
Summary of revisions:
•New optional sixth-order hybrid scheme has been implemented (activated by the flag “hybridScheme” in the input file). The scheme combines the energy-preserving properties of a sixth-order skew-symmetric central difference scheme [1] in smooth flow regions with the shock-capturing properties of a sixth-order targeted essentially non-oscillatory (TENO) scheme at points where shocks are involved. The switch between the two schemes is controlled by a TENO sensor whose cutoff value is adapted based on the maximum value of a modified Ducros sensor [2] across the reconstruction stencil. If the flag “hybridScheme” is set to false, the numerical scheme will revert to the TENO6-A scheme released in the previous version of the solver [3];
•New recycling-rescaling inflow boundary conditions [4,5] for the simulation of turbulent compressible boundary layers are now available to the user;
•Support for Legion tracing, which significantly improves the strong scalability of the solver, has been implemented;
•A diagnostic tool to monitor the time evolution of the flow variables in a subvolume of the computational domain is now available;
•A single-step chemistry mechanism for methane/oxygen combustion has been added to the mixtures handled by the HTR solver;
•Sample scripts for strong scaling have been added to the “testcases” directory;
•Unit test and regression test suites have been added to the repository;
•The input file scheme has been modified in order to reduce verbosity and increase flexibility in specifying the boundary conditions and type of gas mixture;
•Hyperbolic sine stretching functions have been made available to users during the grid generation process;
•Computationally intensive tasks have been ported to C++ and CUDA in order to achieve higher efficiency on all hardware;
•Several optimizations of the tasks body and mapper have been implemented in order to increase the computational efficiency and reduce the memory footprint.
Nature of problem: This code solves the Navier–Stokes equations at hypersonic Mach numbers including finite-rate chemistry for dissociating air and multicomponent transport. The solver is designed for direct numerical simulations (DNS) of transitional and turbulent hypersonic turbulent flows under high-enthalpy conditions, and it accounts for thermochemical effects (vibrational excitation and chemical dissociation).
Solution method: This code uses a low-dissipation sixth-order schemes for the spatial discretization of the conservation equations on Cartesian stretched meshes. Time advancement is carried out by either an explicit method if chemistry is slow, hence not introducing additional stiffness, or by an operator-splitting algorithm whereby chemical production rates are handled implicitly.
Additional comments including restrictions and unusual features: The HTR solver builds on the runtime Legion [6,7] and is written in the programming language Regent [8,9] developed at Stanford University. Instructions for installation of the components are provided in the README file enclosed with the HTR solver and in the Legion repository [6]. References [1] S. Pirozzoli, Journal of Computational Physics 229 (2010) 7180–7190. https://doi.org/10.1016/j.jcp.2010.06.006. F. Ducros, F. Laporte, T. Soulères, V. Guinot, P. Moinat, B. Caruelle, Journal of Computational Physics 161 (2000) 114–139. https://doi.org/10.1006/jcph.2000.6492. M. Di Renzo, L. Fu, J. Urzay, Computer Physics Communications 255 (2020) 107262. https://doi.org/10.1016/j.cpc.2020.107262. T. S. Lund, X. Wu, K. D. Squires, Journal of Computational Physics 140 (1996) 233–258. https://doi.org/10.1006/jcph.1998.5882. S. Pirozzoli, M. Bernardini, F. Grasso, Journal of Fluid Mechanics 657 (2010) 361–393. https://doi.org/10.1017/S0022112010001710. Legion web page, 2020. URL:https://legion.stanford.edu. M. Bauer, S. Treichler, E. Slaughter, A. Aiken, Legion: Expressing locality and independence with logical regions, International Conference for High Performance Computing, Networking, Storage and Analysis, SC (2012), IEEE. Regent web page, 2020. URL:http://regent-lang.org. E. Slaughter, W. Lee, S. Treichler, M. Bauer, A. Aiken, SC ’15: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (2015) 1–12,. https://doi.org/10.1145/2807591.2807629.
In this study, the open-source Hypersonics Task-based Research (HTR) solver for hypersonic aerothermodynamics is described. The physical formulation of the code includes thermochemical effects induced by high temperatures (vibrational excitation and chemical dissociation). The HTR solver uses high-order TENO-based spatial discretization on structured grids and efficient time integrators for stiff systems, is highly scalable in GPU-based supercomputers as a result of its implementation in the Regent/Legion stack, and is designed for direct numerical simulations of canonical hypersonic flows at high Reynolds numbers. The performance of the HTR solver is tested with benchmark cases including inviscid vortex advection, low- and high-speed laminar boundary layers, inviscid one-dimensional compressible flows in shock tubes, supersonic turbulent channel flows, and hypersonic transitional boundary layers of both calorically perfect gases and dissociating air.
Program summary
Program Title: Hypersonics Task-based Research solver
Program Files doi: http://dx.doi.org/10.17632/9zsxjtzfr7.1
Licensing provisions: BSD 2-clause
Programming language: Regent
Nature of problem: This code solves the Navier–Stokes equations at hypersonic Mach numbers including finite-rate chemistry for dissociating air and multicomponent transport. The solver is designed for direct numerical simulations (DNS) of transitional and turbulent hypersonic turbulent flows at high-enthalpies, and accounts for thermochemical effects (vibrational excitation and chemical dissociation).
Solution method: This code uses a low-dissipation sixth-order targeted essentially non-oscillatory (TENO) scheme for the spatial discretization of the conservation equations on Cartesian stretched grids. The time advancement is performed either with an explicit method, when the chemistry is slow and therefore does not introduce additional stiffness in the integration, or with an operator-splitting method that integrates the chemical production rates with an implicit discretization.
Additional comments: The HTR solver builds on the runtime Legion [1] and is written in the programming language Regent [2] recently developed at Stanford University. Instructions for the installation of the components are provided in the README file enclosed with the HTR solver and in the Legion repository [1].
References:
[1] Legion web page: https://legion.stanford.edu
[1] Regent web page: http://regent-lang.org
Hypersonic flows are energetic and result in regions of high temperature, causing internal energy excitation, chemical reactions, ionization, and gas-surface interactions. At typical flight conditions, the rates of these processes are often similar to the rate of fluid motion. Thus, the gas state is out of local thermodynamic equilibrium and must be described by conservation equations for the internal energy and chemical state. Examples illustrate how competition between rates in hypersonic flows can affect aerodynamic performance, convective heating, boundary layer transition, and ablation. The conservation equations are outlined, and the most widely used models for internal energy relaxation, reaction rates, and transport properties are reviewed. Gas-surface boundary conditions are described, including finite-rate catalysis and slip effects. Recent progress in the use of first-principles calculations to understand and quantify critical gas-phase reactions is discussed. An advanced finite-rate carbon ablation model is introduced and is used to illustrate the role of rate processes at hypersonic conditions.
Great efforts have been dedicated during the last decades to the research and development of hypersonic aircrafts that can fly at several times the speed of sound. These aerospace vehicles have revolutionary applications in national security as advanced hypersonic weapons, in space exploration as reusable stages for access to low Earth orbit, and in commercial aviation as fast long-range methods for air transportation of passengers around the globe. This review addresses the topic of supersonic combustion, which represents the central physical process that enables scramjet hypersonic propulsion systems to accelerate aircrafts to ultra-high speeds. The description focuses on recent experimental flights and ground-based research programs and highlights associated fundamental flow physics, subgrid-scale model development, and full-system numerical simulations.
This study investigates control-based forcing methods for incompressible homogeneous-isotropic turbulence forced linearly in physical space which result in constant turbulent kinetic energy, constant turbulent dissipation (also constant enstrophy), or a combination of the two based on a least-squares error minimization. The methods consist of proportional controllers embedded in the forcing coefficients. During the transient, the controllers adjust the forcing coefficients such that the controlled quantity achieves very early a minimal relative error with respect to its target stationary value. Comparisons of these forcing methods are made with the non-controlled approaches of Rosales and Meneveau [“Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties,” Phys. Fluids 17, 095106 (2005)] and Carroll and Blanquart [“A proposed modification to Lundgren’s physical space velocity forcing method for isotropic turbulence,” Phys. Fluids 25, 105114 (2013)], using direct numerical simulations (DNS) and large-eddy simulations(LES). The results indicate that the proposed constant-energetics forcing methods shorten the transient period from a user-defined artificial flow field to Navier-Stokes turbulence while maintaining steadier statistics. Additionally, the proposed method of constant kinetic-energy forcing behaves more robustly in coarse LES when initial conditions are employed that favor the occurrence of subgrid-scale backscatter, whereas the other approaches fail to provide physical turbulent flow fields. For illustration, the proposed forcing methods are applied to dilute particle-laden homogeneous-isotropic turbulent flows; the results serve to highlight the influences of the forcing strategies on the disperse-phase statistics.
Rigorous non-reflective boundary conditions for the multi-component, chemically non-equilibrium reactive flows with real gas effect were derived with the coupling of the species conservations and the fluid equations. The chemically non-equilibrium sound speed was obtained directly from the matrix decomposition of the conservation equations. It was shown that the sound speed of chemically non-equilibrium flow with real gas effect is different from the frozen sound speed of the mixture. The present derivation of non-reflective boundary condition provides the consistency in the characteristic wave speeds for both numerical algorithms of governing equations and boundary conditions. In the limit of single species, the present result reduces to the previous non-reflective boundary conditions given by Poinsot and Lele. The boundary conditions were tested by the shock wave, acoustic wave, vortex and diffusion and premixed flame propagation involving the detailed chemistry and variable thermal properties. The results showed that improper boundary condition generates reflective waves and high frequency noise in the computation domain. It is demonstrated that the present boundary conditions are robust and allow the acoustic and combustion waves propagating out the boundary without causing wave reflection. © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Numerical simulations of axisymmetric flows over reentry configurations at hypersonic conditions using a Navier-Stokes solver are presented. The Navier-Stokes equations are modified using Park's two-temperature model to account for thermochemical nonequilib-rium and weak ionization effects. The finite-volume method is used to solve the set of differential equations. The code has the capability to handle any mixture of hexahedra, tetrahedra, prisms and pyramids in 3D or triangles and quadrilaterals in 2D. The results in this paper only use quadrilaterals. Numerical fluxes between the cells are discretized using a modified Steger-Warming Flux Vector Splitting approach which has low dissipation and is appropriate to calculate boundary layers. A point or line implicit method is used to perform the time integration. Pressure, heat transfer rates and electron number density profiles are compared to available experimental and flight measurements.
The interaction between isotropic turbulence and a normal shock wave is investigated through a series of direct numerical simulations at different Reynolds numbers and mean and turbulent Mach numbers. The computed data are compared to experiments and linear theory, showing that the amplification of turbulence kinetic energy across a shock wave is described well using linearized dynamics. The post-shock anisotropy of the turbulence, however, is qualitatively different from that predicted by linear analysis. The jumps in mean density and pressure are lower than the non-turbulent Rankine–Hugoniot results by a factor of the square of the turbulence intensity. It is shown that the dissipative scales of turbulence return to isotropy within about 10 convected Kolmogorov time scales, a distance that becomes very small at high Reynolds numbers. Special attention is paid to the ‘broken shock’ regime of intense turbulence, where the shock can be locally replaced by smooth compressions. Grid convergence of the probability density function of the shock jumps proves that this effect is physical, and not an artefact of the numerical scheme.
The interaction of a spatially developing adiabatic boundary layer flow at M∞ = 2.25 and Reθ = 3725 with an impinging oblique shock wave (β = 33.2°) is analyzed by means of direct numerical simulation of the compressible Navier-Stokes equations. Under the selected flow conditions the incoming boundary layer undergoes mild separation due to the adverse pressure gradient. Coherent structures are shed near the average separation point and the flow field exhibits large-scale low-frequency unsteadiness. The formation of the mixing layer is primarily responsible for the amplification of turbulence, which relaxes to an equilibrium state past the interaction. Complete equilibrium is attained in the inner part of the boundary layer, while in the outer region the relaxation process is incomplete. Far from the interaction zone, turbulence exhibits a universal behavior and it shows similarities with the incompressible case. The interaction of the coherent structures with the incident shock produces acoustic waves that propagate upstream, thus inducing an oscillatory motion of the separation bubble and a subsequent flapping motion of the reflected shock. The simulation indicates the occurrence of low-frequency tones in the interaction zone associated with peaks in the pressure spectra at discrete frequencies. We propose that such large-scale low-frequency unsteadiness is due to a resonance mechanism that establishes in the interaction region, and which has close similarities with those responsible for the generation of tones in cavity flows and screeching jets. In order to support our claim, we develop a simplified model for the acoustic resonance that is capable to predict the characteristic frequencies of the tones.
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.
A large number of data points for the vibrational relaxation time (pΤv in atm sec) of simple systems have been logarithmically plotted vs (T°K)—⅓. It appears that each system is well represented by a straight line, and that most of these straight lines when extended to higher temperatures intersect near the point [pΤv=10—8 atm sec, (T°K)—⅓=0.03]. Systems with a small reduced mass μ are exceptions to such a simple convergence, and in an improved scheme, the location of the convergence point is dependent on the reduced mass. Such a presentation has lead to an empirical equation correlating available measurements of vibrational relaxation times: log10(pΤv)=(5.0×10−4)μ12θ43[T−13−0.015μ14]−8.00, where θ is the characteristic temperature of the oscillator in K deg. This equation reproduces the measured times within 50% for systems as diverse as N2, I2, and O2&sngbnd;H2. In the worst case thus far, O2&sngbnd;Ar near 1000°K, it is off by a factor of 5.
Improved Navier–Stokes characteristic boundary conditions (NSCBC) are formulated for the direct numerical simulations (DNS) of laminar and turbulent counterflow flame configurations with a compressible flow formulation. The new boundary scheme properly accounts for multi-dimensional flow effects and provides nonreflecting inflow and outflow conditions that maintain the mean imposed velocity and pressure, while substantially eliminating spurious acoustic wave reflections. Applications to various counterflow configurations demonstrate that the proposed boundary conditions yield accurate and robust solutions over a wide range of flow and scalar variables, allowing high fidelity in detailed numerical studies of turbulent counterflow flames.
Procedures to define accurate boundary conditions for reactive flows described by Navier-Stokes equations are discussed. A formulation based on one-dimensional characteristic waves relations at the boundaries, previously developed by Poinsot and Lele for perfect gases with constant homogeneous thermodynamic properties, is rewritten and extended in order to be used in the case of gases described with realistic thermodynamic and reactive models. This kind of formulation appears to be particularly accurate and stable, which is a necessity for non-dissipative codes, in particular for direct simulation of turbulent reactive flows. The simple and solid physical basis of the method is also very attractive and makes it an easy technique to implement in any Navier-Stokes solver. Examples of application in several different computations performed with mixtures of gases and using detailed chemistry and thermodynamic modeling are described. In all cases, acoustic waves, entropy waves, and flames are proved to propagate without perturbation through the boundaries.
The physical properties behind a normal shock in nitrogen are calculated as a function of time. These include the variation of temperature, composition, ionization, and the intensity of radiation from the N2+ first negative band system. This calculation incorporates a rate equation for the dissociation of nitrogen, the conservation laws, an equation describing vibrational relaxation, and a method of coupling the vibrational relaxation with the dissociation rate. The N2+ radiation is computed assuming excitation of the radiating state by collision with vibrationally excited nitrogen molecules. A particular case is considered for which experimental data are available, and regions sensitive to particular rates are indicated.
Equations are derived for the viscosity, ordinary (pressure) diffusion, and thermal diffusion of multicomponent mixtures of gases. The ordinary diffusion velocities are expressed in terms of the usual diffusion coefficients for binary mixtures. The analysis is an extension of the work of Chapman and Cowling. It is shown that the Chapman‐Cowling and Enskog procedure can be justified on the basis of a variational principle. This variational principle should be generally applicable to a large class of problems.
By application of the kinetic theory, with several simplifying assumptions, the previous equation of Buddenberg and the author has been modified to give a general equation for viscosity as a function of molecular weights and viscosities of the pure components of the mixture. Agreement of the equation with experimental data is demonstrated for a number of highly irregular binary gas systems and mixtures of three to seven components.
In this paper we present direct numerical simulations (DNS) of hypersonic turbulent boundary layers to study high-enthalpy effects. We study high- and low-enthalpy conditions, which are representative of those in hypersonic flight and ground-based facilities, respectively. We find that high-enthalpy boundary layers closely resemble those at low enthalpy. Many of the scaling relations for low-enthalpy flows, such as van-Driest transformation for the mean velocity, Morkovin’s scaling and the modified strong Reynolds analogy hold or can be generalized for high-enthalpy flows by removing the calorically perfect-gas assumption. We propose a generalized form of the modified Crocco relation, which relates the mean temperature and mean velocity across a wide range of conditions, including non-adiabatic cold walls and real gas effects. The DNS data predict Reynolds analogy factors in the range of those found in experimental data at low-enthalpy conditions. The gradient transport model approximately holds with turbulent Prandtl number and turbulent Schmidt number of order unity. Direct compressibility effects remain small and insignificant for all enthalpy cases. High-enthalpy effects have no sizable influence on turbulent kinetic energy (TKE) budgets or on the turbulence structure.
As an extension of the authors' work on isotropic vortical
turbulence interacting
with a shock wave (Lee, Lele & Moin 1993), direct numerical simulation
and linear
analysis are performed for stronger shock waves to investigate the effects of the
upstream shock-normal Mach number (M1). A shock-capturing scheme is developed
to accurately simulate the unsteady interaction of turbulence with shock waves.
Turbulence kinetic energy is amplified across the shock wave, and this amplification
tends to saturate beyond M1 = 3.0. An existing controversy between experiments
and theoretical predictions on length scale change is thoroughly investigated through
the shock-capturing simulation: most turbulence length scales decrease across the
shock, while the dissipation length scale
(ρq3/ε) increases slightly for shock
waves with M1<1.65. Fluctuations in thermodynamic
variables behind the shock wave are
nearly isentropic for M1<1.2, and deviate significantly
from isentropy for the stronger
shock waves, due to the entropy fluctuation generated through the interaction.
Direct numerical simulation data of a Mach 2.9, 24○ compression ramp configuration are used to analyse the shock motion. The motion can be observed from the animated DNS data available with the online version of the paper and from wall-pressure and mass-flux signals measured in the free stream. The characteristic low frequency is in the range of (0.007–0.013) U∞/δ, as found previously. The shock motion also exhibits high-frequency, of O(U∞/δ), small-amplitude spanwise wrinkling, which is mainly caused by the spanwise non-uniformity of turbulent structures in the incoming boundary layer. In studying the low-frequency streamwise oscillation, conditional statistics show that there is no significant difference in the properties of the incoming boundary layer when the shock location is upstream or downstream. The spanwise-mean separation point also undergoes a low-frequency motion and is found to be highly correlated with the shock motion. A small correlation is found between the low-momentum structures in the incoming boundary layer and the separation point. Correlations among the spanwise-mean separation point, reattachment point and the shock location indicate that the low-frequency shock unsteadiness is influenced by the downstream flow. Movies are available with the online version of the paper.
Previously developed characteristic-wave-based boundary conditions for multicomponent perfect gas mixtures are here extended to real gas mixtures. The characteristic boundary conditions are derived from the one-dimensional wave decomposition of the Euler equations, and the wave amplitude variations are determined from the prescribed boundary conditions on the flow variables. The viscous conditions are applied separately. For multidimensional simulations, the boundary conditions for each coordinate direction are applied additively. These boundary conditions are tested on a representative two-dimensional problem—the propagation of an incompressible vortex by a supersonic flow with outflow conditions specified as nonreflecting—solved using a high-order finite-difference scheme. Simulations conducted for a heptane–nitrogen mixture flow with strong real gas effects display excellent, nonreflective wave behavior as the vortex leaves the computational domain, verifying the suitability of this method for the multidimensional multicomponent real gas flows computed.
Part I [the author, ibid. 68, 1-24 (1987; Zbl 0619.76089)] introduced the concept of nonreflecting boundary conditions for hyperbolic equations in more than one dimension. This paper develops a general boundary condition formalism for all types of boundary conditions to which hyperbolic systems are subject (including the nonreflecting conditions). The formalism is described in detail, and many examples are provided for common problems in hydrodynamics, including solid wall and nonreflecting boundaries.
Procedures to define boundary conditions for Navier-Stokes equations are discussed. A new formulation using characteristic wave relations through boundaries is derived for the Euler equations and generalized to the Navier-Stokes equations. The emphasis is on deriving boundary conditions compatible with modern non-dissipative algorithms used for direct simulations of turbulent flows. These methods have very low dispersion errors and require precise boundary conditions to avoid numerical instabilities and to control spurious wave reflections at the computational boundaries. The present formulation is an attempt to provide such conditions. Reflecting and non-reflecting boundary condition treatments are presented. Examples of practical implementations for inlet and outlet boundaries as well as slip and no-slip walls are presented. The method applies to subsonic and supersonic flows. It is compared with a reference method based on extrapolation and partial use of Riemann invariants. Test cases described include a ducted shear layer, vortices propagating through boundaries, and Poiseuille flow. Although no mathematical proof of well-posedness is given, the method uses the correct number of boundary conditions required for well-posedness of the Navier-Stokes equations and the examples reveal that it provides a significant improvement over the reference method.
Navier–Stokes characteristic boundary conditions (NSCBC) usually assume the flow to be normal to the boundary plane. In this paper, NSCBC is extended to account for convection and pressure gradients in boundary planes, resulting in a 3D-NSCBC approach. The introduction of these additional transverse terms requires a specific treatment for the computational domain’s edges and corners, as well as a suited set of compatibility conditions for boundaries joining regions associated to different flow properties, as inlet, outlet or wall. A systematic strategy for dealing with edges and corners is derived and compatibility conditions for inlet/outlet and wall/outlet boundaries are proposed. Direct numerical simulation (DNS) tests are carried out on simplified flow configurations at first. 3D-NSCBC brings a drastic reduction of flow distortion and numerical reflection, even in regions of strong transverse convection; the accuracy and convergence rate toward target values of flow quantities is also improved. Then, 3D-NSCBC is used for large-eddy simulation (LES) of a free jet and an impinging round-jet. Edge and corner boundary treatment, combining multidirectional characteristics and compatibility conditions, yields stable and accurate solutions even with mixed boundaries characterized by bad posedness issues (e.g. inlet/outlet). LES confirms the effectiveness of the proposed boundary treatment in reproducing mean flow velocity and turbulent fluctuations up to the computational domain limits.
Time dependent numerical models for hyperbolic systems, such as the fluid dynamics equations, require time dependent boundary conditions when the systems are solved in a finite domain. The “correct” boundary condition depends on the external solution, but for many problems the external solution is not known. In such cases nonreflecting boundary conditions often produce solutions with the desired behavior. This paper extends the concept of nonreflecting boundary conditions to the multidimensional case in non-rectangular coordinate systems. Results are given for several fluid dynamics test problems: the traveling shock wave, shock tube, spherical explosion, and homologous expansion problems in one dimension, and a traveling shock wave moving at a 45° angle with respect to the x axis in two dimensions.
The effects of different methods of approximating multispecies transport phenomena in models of premixed, laminar, steady-state flames have been studied. Five approximation methods that span a wide range of computational complexity are developed. Identical data for individual species properties have been used for each method. Each approximation method is employed in the numerical solution of a set of five H2O2N2 flames. For each flame the computed species and temperature profiles, as well as the computed flame speeds are found to be very nearly independent of the approximation method used. This does not indicate that transport phenomena are unimportant, but rather that the selection of the input values for the individual species transport properties is more important than the selection of the method used to approximate the multispecies transport.Based on these results we have developed a sixth approximation method that is computationally efficient and which provides results extremely close to the most sophisticated and precise method used.
In this paper we extend our earlier work on the efficient implementation of ENO (essentially non-oscillatory) shock-capturing schemes. We provide a new simplified expression for the ENO construction procedure based again on numerical fluxes rather than cell-averages. We also consider two improvements which we label ENO-LLF (local Lax-Friedrichs) and ENO-Roe, which yield sharper shock transitions, improved overall accuracy, for lower computational cost than previous implementation of the ENO schemes. Two methods of sharpening contact discontinuities—the subcell resolution idea of Harten and the artificial compression idea of Yang, which those authors originally used in the cell average framework—are applied to the current ENO schemes using numerical fluxes and TVD Runge-Kutta time discretizations. The implementation for nonlinear systems and multi-dimensions is given. Finally, many numerical examples, including a compressible shock turbulence interaction flow calculation, are presented.
The rate of dissociation behind strong shock waves in N2 and O2 is calculated using a revised model for the coupling of vibration and dissociation. In previous calculations a model which coupled the rate of dissociation to the degree of vibrational excitation was used. The present model adds to the coupled vibration-dissociation model the fact that the rate of vibrational excitation is in part determined by the rate of dissociation. Since the average energy of molecules which are dissociated is greater than the average energy of the remaining molecules, this coupling results in a drain on the average vibrational energy. It is shown that this coupling reduces the strong overshoot in vibrational energy that was previously obtained, and decreases the rate of dissociation behind strong shocks.
The rate of molecular dissociation behind strong shock waves is calculated with the assumption that dissociation can occur preferentially from the higher vibrational levels. An exponential probability of dissociation from the various vibrational levels is employed using an anharmonic oscillator model. Results for the dissociation of oxygen in an argon diluent are presented. Vibrational non-equilibrium introduces a T−3 temperature dependence into the oxygen dissociation rate constant in the range 4000°–8000°K. A dissociation lag-time of the order of the extrapolated vibrational relax ation time is predicted immediately behind the shock front. The computed results are shown to be in agreement with available experimental results.
The conservation equations for simulating hypersonic flows in thermal and chemical nonequilibrium and details of the associated physical models are presented. These details include the curve fits used for defining thermodynamic properties of the 11 species air model, curve fits for collision cross sections, expressions for transport properties, the chemical kinetics models, and the vibrational and electronic energy relaxation models. The expressions are formulated in the context of either a two or three temperature model. Greater emphasis is placed on the two temperature model in which it is assumed that the translational and rotational energy models are in equilibrium at the translational temperature, T, and the vibrational, electronic, and electron translational energy modes are in equilibrium at the vibrational temperature, T sub v. The eigenvalues and eigenvectors associated with the Jacobian of the flux vector are also presented in order to accommodate the upwind based numerical solutions of the complete equation set.
The existing experimental data on the rate coefficients for the chemical reactions in nonequilibrium high temperature air are reviewed and collated, and a selected set of such values is recommended for use in hypersonic flow calculations. For the reactions of neutral species, the recommended values are chosen from the experimental data that existed mostly prior to 1970, and are slightly different from those used previously. For the reactions involving ions, the recommended rate coefficients are newly chosen from the experimental data obtained more recently. The reacting environment is assumed to lack thermal equilibrium, and the rate coefficients are expressed as a function of the controlling temperature, incorporating the recent multitemperature reaction concept.
A number of chemical-kinetic problems related to phenomena occurring behind a shock wave surrounding an object flying in the earth atmosphere are discussed, including the nonequilibrium thermochemical relaxation phenomena occurring behind a shock wave surrounding the flying object, problems related to aerobraking maneuver, the radiation phenomena for shock velocities of up to 12 km/sec, and the determination of rate coefficients for ionization reactions and associated electron-impact ionization reactions. Results of experiments are presented in form of graphs and tables, giving data on the reaction rate coefficients for air, the ionization distances, thermodynamic properties behind a shock wave, radiative heat flux calculations, Damkoehler numbers for the ablation-product layer, together with conclusions.