Compressible Direct Numerical Simulation Research Articles

EFFECTS OF LOCALIZED NON-GAUSSIAN ROUGHNESS ON HIGH PRESSURE TURBINE AERO-THERMAL PERFORMANCE: CONVECTIVE HEAT TRANSFER, SKIN-FRICTION AND THE REYNOLDS' ANALOGY

Abstract Compressible direct numerical simulations are conducted to investigate how surface roughness affects the aero-thermal performance of a high pressure turbine vane operating at an exit Reynolds number of 0.59 × 106 and exit Mach number of 0.92. The roughness under investigation here was synthesized with non- Gaussian statistical properties and an amplitude that varies over its chord length — representative of what truly occurs on an in-service vane. Particular attention is directed towards how sys- tematically changing the axial extent of leading edge roughness affects convective heat transfer (Nusselt and Stanton numbers) and aerodynamic drag (skin-friction coefficient) on the pressure and suction surfaces. The results of this investigation demonstrate that moving the larger amplitude roughness further along the suc- tion surface can cause major changes in the blade boundary-layer state. In fact, towards the trailing-edge of one of the rough vanes investigated here, the local skin-friction coefficient increases by a factor of twenty-three (2, 300%) compared to smooth-vane lev- els, whereas the local Stanton number increases by a factor six (600%). The disproportionate rise of drag compared to heat transfer is explored in further detail by quantifying the Reynolds' analogy and by calculating the fractional contributions of pres- sure drag and viscous drag to the total drag force. The effect of varying the inlet turbulence intensity and integral length-scale for a fixed roughness topography is also investigated, along with the Reynolds-number scaling of heat transfer and drag.

A particle-resolved direct numerical simulation method for the compressible gas flow and arbitrary shape solid moving with a uniform framework

Compressible particle-resolved direct numerical simulations (PR-DNS) are widely used in explosion-driven dispersion of particles simulations, multiphase turbulence modelling, and stage separation for two-stage-to-orbit vehicles. The direct forcing immersed boundary method (IBM) is a promising method and widely applied in low speed flow while there is few research regarding compressible flows. We developed a novel IBM to resolve supersonic and hypersonic gas flows interacting with irregularly shaped multi-body particle. The main innovation is that current method can solve the interaction of particles and high-speed fluids, particle translation and rotation, and collision among complex-shaped particles within a uniform framework. Specially, high conservation and computation consumption are strictly satisfied, which is critical for resolving the high speed compressible flow feature. To avoid the non-physical flow penetration around particle surface, an special iterative algorithm is specially derived to handle the coupling force between the gas and particles. The magnitude of the velocity difference error could be reduced by 6–8 orders compared to that of a previous method. Additionally, aerodynamic force integration was achieved using the momentum equation to ensure momentum conservation for two-phase coupling. A high-efficiency cell-type identification method for each step was proposed, and mapping among LPs and cells was used again to select the immersed cells. As for the collision force calculation, the complex shape of a particle was represented by a cloud of LPs and the mapping of LPs and cells was used to reduce the complexity of the algorithm for contact searching. The repetitive use of the mapping relationship could reduce the internal memory and improve the efficiency of the proposed algorithm. Moreover, various verification cases were conducted to evaluate the simulation performance of the proposed algorithm, including two- and three-dimensional moving and motionless particles with regular and complex shapes interacting with high-speed flow. Specifically, an experiment involving a shock passing through a sphere was designed and conducted to provide high-precision data. The corresponding results of the large-scale numerical simulation agree well with those obtained experimentally. The current method supports flow simulations at a particle-resolved scale in engineering.

Subconvective wall-pressure fluctuations in low-Mach-number turbulent channel flow

Compressible direct numerical simulations are employed to elucidate the low-wavenumber behaviour of wall-pressure fluctuations in turbulent channel flow and the effect of flow Mach number in the nearly incompressible regime. Simulations are conducted at bulk Mach numbers 0.4, 0.2 and 0.1, and friction Reynolds number 180. In addition to the convective ridge that is virtually Mach-number-independent, acoustic ridges, whose magnitudes are orders of magnitude lower, are identified in the two-dimensional wavenumber–frequency spectrum. At lower frequencies, the acoustic ridges represent propagating longitudinal and oblique waves that match the theoretical predictions of two-dimensional duct modes with a uniform mean flow. They decay with decreasing Mach number but remain distinctly identifiable even at Mach 0.1. At high frequencies, in contrast, no propagating waves are found, and the spectral level in the supersonic wavenumber range is broadly elevated and increases with decreasing Mach number.

Opposite effects of flame-generated potential and solenoidal velocity fluctuations on flame surface area in moderately intense turbulence

To explore the influence of dilatational and rotational motions on iso-scalar surface area within a premixed turbulent flame, three-dimensional compressible direct numerical simulation data obtained by Dave et al. (2018) from a lean, complex-chemistry, hydrogen-air flame propagating in intense small-scale turbulence (a ratio of laminar flame thickness to Kolmogorov length scale is about 20) in a box are analyzed using Helmholtz–Hodge decomposition of velocity field into solenoidal and potential components. The obtained results indicate that the flame surface area is created by potential velocity fluctuations generated due to combustion-induced thermal expansion, whereas the rotational motion acts to smooth wrinkles of iso-scalar surfaces within local preheat and reaction zones and, consequently, to reduce the flame surface area. The latter finding challenges the classical concept of flame-generated turbulence and is attributed to flame-generated solenoidal velocity fluctuations. Specifically, the incoming tangential (to the flame) vorticity is suppressed by baroclinic torque, which also generates vorticity in another tangential direction, with the latter (flame-generated) solenoidal velocity fluctuations working to smooth the flame surface. Thus, even under conditions of moderately intense turbulence, flame surface area can primarily be created by potential velocity perturbations caused by combustion-induced thermal expansion.

Wall-pressure fluctuations in weakly compressible turbulent channel flow

Wall-pressure fluctuations in turbulent wall-bounded flows are detrimental in many applications because they can cause structural vibrations and acoustic radiation. Their spectral behavior at subconvective wavenumbers are to date poorly understood and predicted, particularly in low-Mach-number flows. In this study, compressible direct numerical simulation is employed to elucidate the low-wavenumber behavior of wall-pressure fluctuations in turbulent channel flow and the effect of flow Mach number in the nearly incompressible regime. Simulations are conducted at bulk Mach numbers of 0.4, 0.2, and 0.1, and friction Reynolds number of 180. In addition to the convective ridge that is virtually Mach-number independent, acoustic ridges representing longitudinal and oblique waves are clearly identified in the two-dimensional wavenumber-frequency spectrum. The acoustic peaks are orders of magnitude weaker than the convective peak and decay with flow Mach number, but remain distinctly identifiable even at Mach 0.1. The acoustic energy in the supersonic wavenumber range is significantly enhanced by the onset of the first oblique mode but not much affected by the higher modes. The effect of a small, two-dimensional surface hump is also considered, which is shown to elevate the spectral level of the fluctuating wall pressure in the subconvective wavenumber range by several decades due to acoustic diffraction by the hump. [Work supported by the U.S. Office of Naval Research.]

Investigation of strong isothermal stratification effects on multi-mode compressible Rayleigh–Taylor instability

Rayleigh–Taylor instability, RTI, occurs at the interface separating two fluids subjected to acceleration when the density gradient and the acceleration are in opposite directions. Previous scientific research primarily considered RTI under the incompressible assumption, which may not be valid in many high-energy-density engineering applications and astrophysical phenomena. In this study, the compressibility effects of the background isothermal stratification strength on multi-mode two-dimensional RTI are explored using fully compressible multi-species direct numerical simulations. Cases under three different isothermal Mach numbers – Ma=0.15, 0.3, and 0.45 – are investigated to explore weakly, moderately, and strongly stratified compressible RTI, respectively, at an Atwood number of 0.04. Unlike incompressible RTI, an increase in the flow compressibility through the strength of the background stratification can suppress the RTI growth and can lead to a termination of the RTI mixing layer growth with a highly molecularly mixed state. Our findings suggest that even at the chosen relatively low Atwood number, the variable-density effects can be significantly enhanced due to an increase in the background stratification for the compressible RTI as different spatial profiles become noticeably asymmetric across the mixing layer for the strongly stratified case. In addition, this study compares the chaotic behavior of the cases by studying the transport of the turbulent kinetic energy as well as the vortex dynamics. The Reynolds number dependence of the results is also examined with three different Reynolds numbers, and the findings for the large-scale mixing and flow quantities of interest are shown to be universal in the range of the Reynolds numbers studied.

Turbulence and added drag over acoustic liners

We present pore-resolved compressible direct numerical simulations of turbulent flows grazing over perforated plates, that closely resemble the acoustic liners found on aircraft engines. Our direct numerical simulations explore a large parameter space including the effects of porosity, thickness and viscous-scaled diameter of the perforated plates, at friction Reynolds numbers $\textit {Re}_\tau = 500\unicode{x2013}2000$ , which allows us to develop a robust theory for estimating the added drag induced by acoustic liners. We find that acoustic liners can be regarded as porous surfaces with a wall-normal permeability and that the relevant length scale characterizing their added drag is the inverse of the wall-normal Forchheimer coefficient. Unlike other types of porous surfaces featuring Darcian velocities inside the pores, the flow inside the orifices of acoustic liners is fully turbulent, with a magnitude of the wall-normal velocity fluctuations comparable to the peak in the near-wall cycle. We provide clear evidence of a fully rough regime for acoustic liners, also confirmed by the increasing relevance of pressure drag. Once the fully rough asymptote is reached, canonical acoustic liners provide an added drag comparable to that of sand-grain roughness with viscous-scaled height matching the inverse of the viscous-scaled Forchheimer permeability of the plate.

Direct numerical simulation of vertically heated natural convection over 3D irregular roughness

Natural convection in a cavity having an aspect ratio of 4 with irregular roughness on vertical isothermal sidewalls is investigated through compressible direct numerical simulation at a Rayleigh number of 1 × 1010. The compressible solver combined with the immersed boundary adopted in this paper is validated. The profiles of probability density functions of the thermal and hydrodynamic quantities show that at this Rayleigh number, the roughness increases the instability of the nearby fluid. In the instantaneous results, the isosurfaces of the Q-criterion reveal that a vortex is generated near roughness peaks and increases the local heat transfer, but in the valley regions, it is difficult to find vortex structures like in the peak regions. Results for the quasi-steady state show that the cavity with rough sidewalls has a lower Nusselt number on the hot wall than the smooth cavity. In other words, the roughness adversely affects the wall heat transfer. This indicates that the hot sidewall with roughness conveys less energy than the smooth wall. As a consequence, the fluid in the cavity with rough sidewalls has lower wall shear stress than that with smooth sidewalls. However, the results of the eddy heat flux show that turbulent flows generated by roughness enhance the heat transfer near the sidewalls. As the mixing effects of turbulent flows strengthen, the disparity of the average Nusselt number at the sidewall between rough and smooth cases reduces.

Adverse-pressure-gradient turbulent boundary layer on convex wall

Direct numerical simulations (DNSs) of an incompressible turbulent boundary layer on an airfoil (suction side) and that on a flat plate are compared to characterize the non-equilibrium turbulence and the effect of wall curvature on the flow. The two simulations effectively impose matching streamwise distributions of adverse pressure gradient (APG) quantified by the acceleration parameter (K). For the airfoil flow, an existing compressible DNS carried out by Wu et al. [“Effects of pressure gradient on the evolution of velocity-gradient tensor invariant dynamics on a controlled-diffusion aerofoil at Rec = 150,000,” J. Fluid Mech. 868, 584–610 (2019)] of the flow around a controlled-diffusion airfoil is used. For the flat-plate flow, a separate simulation is carried out with the aim to reproduce the flow in the region of the airfoil boundary layer with zero to adverse pressure gradients. Comparison between the two cases extracts the effect of a mild convex wall curvature on velocity and wall-pressure statistics in the presence of APG. In the majority part of the boundary layer development, curvature effect on the flow is masked by that of the APG, except for the region with weak pressure gradients or a thick boundary layer where the effect of wall curvature appears to interact with that of APG. High-frequency wall-pressure fluctuations are also augmented by the wall curvature. Overall, the boundary layers are qualitatively similar with and without the wall curvature. This indicates that a flat-plate boundary layer DNS may serve as a low-cost surrogate of a boundary layer over the airfoil or other objects with mild curvatures to capture important flow features to aid modeling efforts.

A numerical study on the combustion of a resolved carbon particle

Combustion of a single, resolved carbon particle is studied using a novel numerical approach that makes use of an overset grid. The model is implemented into the framework of a compressible Direct Numerical Simulation (DNS) code. A method to artificially reduce the speed of sound is presented. For Mach numbers lower than ∼0.1 this method may dramatically improve numerical efficiency without affecting any physical aspects except for the acoustics. The ability of the model to simulate solid fuel combustion is demonstrated and all parts of the model are validated against experimental and numerical data. A sensitivity of the carbon conversion rate to selected parameters (diffusion coefficients and homogeneous and heterogeneous kinetics) is investigated. A strong dependence on the oxygen diffusivity is observed and explained.

Response of Autoignition-Stabilized Flames to One-Dimensional Disturbances: Intrinsic Response

Abstract Burning carbon-free fuels such as hydrogen in gas turbines promises power generation with reduced greenhouse gas emissions. A two-stage combustor architecture with an autoignition-stabilized flame in the second stage allows for efficient combustion of hydrogen fuels. However, interactions between the autoignition-stabilized flame and the acoustic field of the combustor may result in self-sustained oscillations of the flame front position and heat release rate, which severely affect the stable operation of the combustor. We study one such 'intrinsic' mode of interaction wherein acoustic waves generated by the unsteady flame travel upstream and modulate the incoming mixture resulting in flame front oscillations. In particular, we study the response of an autoignition-stabilized flame to upstream traveling acoustic disturbances in a one-dimensional configuration. We first present a numerical framework to calculate the response of autoignition-stabilized flames to acoustic and entropy disturbances in a one-dimensional combustor. The flame response is computed by solving the energy and species mass balance equations. We validate the framework with compressible direct numerical simulations. Lastly, we present results for the flame response to upstream traveling acoustic perturbations. The results show that autoignition-stabilized flames are highly sensitive to acoustic temperature fluctuations and exhibit a characteristic frequency-dependent response. Acoustic pressure and velocity fluctuations constructively or destructively superpose with temperature fluctuations, depending on the mean pressure and relative phase between the fluctuations. The findings of the present work are essential for understanding the intrinsic feedback mechanism in combustors with autoignition-stabilized flames.

Numerical study on edge tone with compressible direct numerical simulation: Sound intensity and jet motion

A two-dimensional model of the edge tone is studied by a highly accurate and reliable method of direct numerical simulation of the compressible Navier-Stokes equations, and used to verify key features observed in previous experimental and numerical studies, and to discover new features related to the jet motion and the edge tone generation mechanism. The first and second modes of the edge tone that are numerically reproduced agree well with Brown’s equation. In the mode transition region, dynamical mode transition is observed at a fixed jet velocity. For both first and second modes, the pressure distributions are antisymmetric with respect to the edge plate, and the sound intensity is proportional to the fifth power of the jet velocity. These results are consistent with the edge tone being radiated from a dipole-like source. Spatial profiles of the velocity and the velocity variance of the oscillating jet are also investigated for each mode over a range of the jet velocity including the mode transition regime. The amplitude of the velocity oscillation becomes constant with increasing jet velocity, while a measure of the amplitude of the velocity variance profile, which is introduced to characterize the strength of the jet fluctuation and named the ’fluctuation strength’, is proportional to the third power of the jet velocity. Some properties of the fluctuation strength correspond to properties of the sound intensity, including the first mode having larger amplitude than the second mode, and the way of deviating from the power law at smaller values of jet velocity and in the mode transition region. It is proposed that the third-power law exhibited by behavior of the fluctuation strength could be related to the increase of the skewness observed in the velocity profile with increase of jet velocity, and a model calculation is used to support this proposal.

Development and validation study of a 1D analytical model for the response of reheat flames to entropy waves

Numerical simulations of laminar premixed flames burning hydrogen and methane in spontaneous ignition mode are performed by harmonically exciting the reactants’ temperature at the domain inlet. The results are compared to an analytical model representing the same reactive flow configuration. The model provides a simplified but nevertheless accurate representation of reheat combustion taking place in sequential gas turbine combustors. An analytic expression for autoignition flames transfer functions to entropy waves is derived and used to extend transfer function models from the literature. For validation purposes, results from fully compressible Direct Numerical Simulations (DNS), including a complete representation of the fluctuating acoustic and entropic fields of the reactive flow, are analyzed and compared to incompressible Unsteady Reynolds-Averaged Navier–Stokes (URANS) simulations that only take into account the fluctuating entropic field. Methane flames are found to be more sensitive to entropic forcing than hydrogen flames, featuring nonlinear phenomena even for low excitation amplitudes. In the linear regime, all flames behave as predicted by the analytical model and the URANS simulations are found to correctly predict the fluctuating entropic field. The transition from linear to nonlinear flame response is described in detail and its physical mechanisms are explained. Comparisons with results available in the literature show good prediction capabilities, both in terms of flame describing function and integrated heat release rate. Limitations of the proposed analytical model with respect to real combustion systems are discussed and a simple correction is proposed.

Effects of Biogas Composition on the Edge Flame Propagation in Igniting Turbulent Mixing Layers

Three-dimensional compressible Direct Numerical Simulations have been used to investigate the localised forced ignition of statistically planar biogas/air mixing layers for different levels of turbulence intensity and biogas composition. The biogas is represented by a hbox {CH}_4/hbox {CO}_2 mixture and a two-step mechanism capturing the variation of the unstrained laminar flame speed with equivalence ratio and hbox {CO}_2 dilution was used. The mixture composition was found to significantly affect the flame kernel development which was reflected in the diminished growth rate of the burned gas volume with increasing hbox {CO}_2 dilution. A successful ignition of hbox {CH}_4/hbox {CO}_2/air mixing layer gives rise to a tribrachial flame structure involving fuel-rich and lean premixed branches on either side of the diffusion flame stabilised on the stoichiometric mixture fraction iso-surface. The most probable edge flame speed decreases in time and converges to a value that is at most equal to its laminar theoretical limit, and can even locally become negative for large values of the dilution and/or turbulence intensity. The decomposition of the edge flame speed showed a negligible or negative contribution of the mixture fraction surface displacement speed, while the displacement speed of the fuel mass fraction surface appeared as the dominant contributor. Finally, the edge flame speed dependences to the fuel mass fraction and mixture fraction gradients, fuel mass fraction iso-surface curvature and tangential strain rate have been analysed and found, within the dilution values considered, qualitatively similar to those of undiluted mixtures regardless of the amount of hbox {CO}_2, although quantitative differences were observed.

On the noise generated by a controlled-diffusion aerofoil at [formula omitted

A compressible direct numerical simulation of the flow and near-field acoustics over a Controlled-Diffusion aerofoil at 8∘ geometrical angle of attack, a Reynolds number of Rec=1.5×105 based on the chord and a free-stream Mach number of M=0.25, with a spanwise extent of 12% chord was conducted to produce a space-time database to determine the aerofoil noise generation mechanisms. Three main noise sources respectively from the flow separation and reattachment at the leading-edge on the suction side, the interaction between the attached turbulent flow on the suction side and the trailing-edge and an additional near-wake source due to secondary instability are observed. It should be noticed, however, that such an additional source may strongly depend on the aerofoil shape and also on the flow conditions. The contribution of these three noise sources to the noise radiation at different frequencies is analysed through a wavenumber-frequency filtering procedure. In the low to mid frequency range, the contribution of the traditional trailing-edge noise is significant, whereas at high frequencies, the noise radiation is shown to be related to the flow transition/reattachment in the leading-edge area and the flow in the near wake further downstream of the trailing-edge noise source center. The near-wake source is observed to dominate the high-frequency noise radiation.

Direct Numerical Simulation of the Richtmyer–Meshkov Instability in Reactive and Nonreactive Flows

ABSTRACT Uncontrolled hydrogen/air explosions pose a central problem in nuclear and process plant safety research. The Richtmyer–Meshkov Instability (RMI) can be an important contributing factor to flame acceleration and subsequently the deflagration-to-detonation transition. In this context, the RMI is caused by the interaction of a sharp pressure gradient, generated by a shock wave and the density gradient at a flame surface. This interaction leads to the production of baroclinic torque at the flame surface, which causes flame wrinkling. For this work, compressible direct numerical simulations of a shock wave, interacting with a perturbed statistically planar flame in a premixed medium, are conducted. After the first interaction with the flame, a second instance of shock–flame interaction (re-shock) is observed, caused by the reflection of the shock wave from an adiabatic wall on the right-hand side of the domain. The influence of the chemical reaction, shock strength, as well as initial flame surface disturbance on the flame surface area and mixing width , are investigated. It is found that the chemical reaction has a large impact on the development of and , as it partly smoothens emerging wrinkled structures on the flame surface for the present thermochemistry. The maximum reached is hereby reduced by about compared to cases without chemical reaction. A comparison of the development of over time with predictions from an analytical model shows good agreement for the linear portion of the instability (short time frame after shock interaction). The model then fails to predict the decrease of in the non-linear part of the model, due to the smoothing effects of the chemical reaction. The influence of the initial flame disturbance on the development of the flame surface area and mixing width shows a strong dependence on the selected disturbance wavenumber . The maximum achievable flame surface area is inversely proportional to in the investigated range. Increasing the shock Mach number and therefore increasing the pressure gradient showed a strong impact on the development of the RMI. After the first shock interaction, numerous fresh gas funnels are created, reaching into the burned gas. Therefore, when the re-shock interaction occurs, the shock will interact with an increased and more irregular flame surface. In addition, the flame thickness is reduced by about upon each shock interaction, due to flame compression and increase in the pressure level from the shock wave. Both effects will lead to distinct spikes in the baroclinic torque production over time upon flame interaction with the re-shock and result in a strong increase of and .

Trailing-edge broadband noise prediction of an airfoil with boundary-layer tripping

Hybrid methods for trailing edge noise prediction are applied to a NACA 6512-63 airfoil at zero angle of attack embedded in the actual narrow stream of the Siegen open-jet anechoic wind-tunnel. Incompressible large-eddy simulations (LES) of the airfoil trailing-edge flow yield the noise sources, and different analytical acoustic analogies then predict the acoustic far-field pressure. The chord-based Reynolds number of 1.9 × 105 triggers long laminar flow regions along the airfoil, leading to instability waves with an associated broadband noise radiation. To avoid this laminar-instability noise, the airfoil boundary layer is experimentally tripped on both sides. The actual serration tripping is included in the numerical grid as well as a simplified stair-step trip. Aerodynamic and acoustic results of the performed LES are compared with measurements and compressible direct numerical simulation (DNS) results. No noticeable influence of the trip geometry is found whereas the numerical schemes play a more important role. LES predictions with boundary layer trip compare well with experimental measurements and DNS cases. If the boundary-layer trip is not sufficiently thick in the numerical model, or not present at all, the far-field acoustic pressure is overpredicted and resembles more closely the sound radiation from the untripped airfoil with laminar instability noise.

Compressible plane turbulent wakes under pressure gradients evolving in a constant area section

Development of statistically two-dimensional (2-D) turbulent wakes under pressure gradients is a common feature of many industrial and aerodynamic flows. The usual set-up is to generate a wake, and study its development through a passage with a variable area; if the downstream area decreases (increases) a favourable pressure gradient or FPG (adverse pressure gradient or APG) is imposed. In applications such as in turbomachinery, however, the wakes develop in a periodic constant area passage in the stator–rotor gap and with an imposed pressure gradient. To study these flows, here, we develop a canonical set-up for this new kind of wake evolution in FPG and APG of different strengths by placing a 2-D flat plate normal to the flow in a periodic constant area passage and a fixed inflow mass flux. Employing compressible direct numerical simulations, we impose pressure gradients through a ramped body force term to the momentum and total energy equations while the wake is allowed to develop spatially in a region of fixed width. The resultant mean velocity statistics, wake width, energy budgets and entropy generation rates are scrutinised to assess the effect of the pressure gradients, and where possible, the similarities and differences to the conventional case of variable area pressure gradients are discussed. The results show that the effect of a constant area pressure gradient on flow statistics is non-trivial, resulting from significant density changes. The pressure gradients also have an effect on the different energy budgets, which produces a gain for FPG and loss for APG in the mean kinetic energy. Consequently, the entropy generation rate diminishes and augments for the FPG and APG, respectively, compared to the zero pressure gradient. Finally, the effect of different passage heights ( ) relative to the wake half-width ( ) is studied using large eddy simulations. We find that wake width and hence the spreading, depends primarily on the wake–wake interaction for small and pressure gradients for larger , and this has implications for the design of turbomachinery.

Skew-symmetric splitting of high-order central schemes with nonlinear filters for computational aeroacoustics turbulence with shocks

A class of high-order nonlinear filter schemes by Yee et al. (J Comput Phys 150:199–238, 1999), Sjogreen and Yee (J Comput Phys 225:910–934, 2007), and Kotov et al. (Commun Comput Phys 19:273–300, 2016; J Comput Phys 307:189–202, 2016) is examined for long-time integrations of computational aeroacoustics (CAA) turbulence applications. This class of schemes was designed for an improved nonlinear stability and accuracy for long-time integration of compressible direct numerical simulation and large eddy simulation computations for both shock-free turbulence and turbulence with shocks. They are based on the skew-symmetric splitting version of the high-order central base scheme in conjunction with adaptive low-dissipation control via a nonlinear filter step to help with stability and accuracy capturing at shock-free regions as well as in the vicinity of discontinuities. The central dispersion-relation-preserving schemes as well as classical central schemes of arbitrary orders fit into the framework of skew-symmetric splitting of the inviscid flux derivatives. Numerical experiments on CAA turbulence test cases are validated.

Self-similar compressible turbulent boundary layers with pressure gradients. Part 2. Self-similarity analysis of the outer layer

A thorough self-similarity analysis is presented to investigate the properties of self-similarity for the outer layer of compressible turbulent boundary layers. The results are validated using the compressible and quasi-incompressible direct numerical simulation (DNS) data shown and discussed in the first part of this study; see Wenzelet al. (J. Fluid Mech., vol. 880, 2019, pp. 239–283). The analysis is carried out for a general set of characteristic scales, and conditions are derived which have to be fulfilled by these sets in case of self-similarity. To evaluate the main findings derived, four sets of characteristic scales are proposed and tested. These represent compressible extensions of the incompressible edge scaling, friction scaling, Zagarola–Smits scaling and a newly defined Rotta–Clauser scaling. Their scaling success is assessed by checking the collapse of flow-field profiles extracted at various streamwise positions, being normalized by the respective scales. For a good set of scales, most conditions derived in the analysis are fulfilled. As suggested by the data investigated, approximate self-similarity can be achieved for the mean-flow distributions of the velocity, mass flux and total enthalpy and the turbulent terms. Self-similarity thus can be stated to be achievable to a very high degree in the compressible regime. Revealed by the analysis and confirmed by the DNS data, this state is predicted by the compressible pressure-gradient boundary-layer growth parameter$\unicode[STIX]{x1D6EC}_{c}$, which is similar to the incompressible one found by related incompressible studies. Using appropriate adaption,$\unicode[STIX]{x1D6EC}_{c}$values become comparable for compressible and incompressible pressure-gradient cases with similar wall-normal shear-stress distributions. The Rotta–Clauser parameter in its traditional form$\unicode[STIX]{x1D6FD}_{K}=(\unicode[STIX]{x1D6FF}_{K}^{\ast }/\bar{\unicode[STIX]{x1D70F}}_{w})(\text{d}p_{e}/\text{d}x)$with the kinematic (incompressible) displacement thickness$\unicode[STIX]{x1D6FF}_{K}^{\ast }$is shown to be a valid parameter of the form$\unicode[STIX]{x1D6EC}_{c}$and hence still is a good indicator for equilibrium flow in the compressible regime at the finite Reynolds numbers considered. Furthermore, the analysis reveals that the often neglected derivative of the length scale,$\text{d}L_{0}/\text{d}x$, can be incorporated, which was found to have an important influence on the scaling success of common ‘low-Reynolds-number’ DNS data; this holds for both incompressible and compressible flow. Especially for the scaling of the$\bar{\unicode[STIX]{x1D70C}}\widetilde{u^{\prime \prime }v^{\prime \prime }}$stress and thus also the wall shear stress$\bar{\unicode[STIX]{x1D70F}}_{w}$, the inclusion of$\text{d}L_{0}/\text{d}x$leads to palpable improvements.