Supersonic Turbulent Boundary Layer Research Articles

Laws of the wall for velocity and temperature in supersonic turbulent boundary layers

AbstractScaling laws for velocity and temperature profiles in the near‐wall region of sub‐ and supersonic turbulent boundary layers have been developed, which allow us to represent velocity and temperature profiles in compressible gas stream in terms of those in an incompressible boundary layer. They are obtained as asymptotic expansions of the solutions to the Reynolds equations in a small parameter — the Mach number based on the friction velocity and gas enthalpy on the wall. The leading term of the expansion for velocity corresponds to known Van Driest's formula. However, the obtained solution contains additional terms of order unity, which explains the contradiction between Van Driest's formula and experimental data. The law of the wall for temperature, which has been formulated for the first time, has an analogous structure. Besides the von Kármán constant and the turbulent Prandtl number in the logarithmic region, known for incompressible flow, the obtained relations contain three new universal constants, which do not depend on gas molecular properties and the specific heat ratio. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Direct numerical simulation of supersonic turbulent boundary layer subjected to a curved compression ramp

Numerical investigations on a supersonic turbulent boundary layer over a longitudinal curved compression ramp are conducted using direct numerical simulation for a free stream Mach number M∞ = 2.9 and Reynolds number Reθ = 2300. The total turning angle is 24°, and the concave curvature radius is 15 times the thickness of the incoming turbulent boundary layer. Under the selected conditions, the shock foot is transferred to a fan of the compression wave because of the weaker adverse pressure gradient. The time-averaged flow-field in the curved ramp is statistically attached where the instantaneous flow-field is close to the intermittent transitory detachment state. Studies on coherent vortex structures have shown that large-scale vortex packets are enhanced significantly when the concave curvature is aligned in the spanwise direction. Consistent with findings of previous experiments, the effect of the concave curvature on the logarithmic region of the mean velocity profiles is found to be small. The intensity of the turbulent fluctuations is amplified across the curved ramp. Based on the analysis of the Reynolds stress anisotropy tensor, the evolutions of the turbulence state in the inner and outer layers of the boundary layer are considerably different. The curvature effect on the transport mechanism of the turbulent kinetic energy is studied using the balance analysis of the contributing terms in the transport equation. Furthermore, the Görtler instability in the curved ramp is quantitatively analyzed using a stability criterion. The instantaneous streamwise vorticity confirms the existence of the Görtler-like structures. These structures are characterized by an unsteady motion. In addition, the dynamic mode decomposition analysis of the instantaneous flow field at the spanwise/wall-normal plane reveals that four dynamical relevant modes with performance loss of 16% provide an optimal low-order representation of the essential characteristics of the numerical data. The spatial structures of the dominated low-frequency dynamic modes are found to be similar to that of the Görtler-like vortices.

The impact of streamwise convex curvature on the supersonic turbulent boundary layer

With an inflow Mach number of 2.95, the isolated impacts of the convex curvature and the streamwise favorable pressure gradient on the supersonic turbulent boundary layer are experimentally investigated by employing particle image velocimetry. An experiment on the convex turbulent boundary layer with a zero streamwise pressure gradient is carefully arranged to investigate the impact of the streamline curvature. The results are compared with the favorable-pressure-gradient case (with pressure gradient parameter β = −0.60). For both of the zero-pressure-gradient and the favorable-pressure-gradient convex boundary layers, the log law is found to be well preserved in the streamwise velocity profile with inner scaling. The streamline convex curvature and the streamwise favorable pressure gradient are found to have similar impacts on the distribution of mean streamwise velocity. They both weaken the boundary layer’s wake strength, as well as the principal strain rate in the near wall region. While both of the convex streamline curvature and the favorable pressure gradient are observed to dampen the streamwise turbulent fluctuation, their effects on the normal fluctuation are opposite. The convex curvature reinforces the normal fluctuation, and its strengthening effect is much stronger than the relaxation effect of the favorable pressure gradient. Retrograde vortices are notably weakened by the favorable pressure gradient, which contributes to the relaxation of turbulence. The impacts of the streamline convex curvature and the streamwise favorable pressure gradient are more notable in the outer layer, while in the near wall region their contributions are weakened.

Large eddy simulation of cavity-stabilized hydrogen combustion in a diverging supersonic combustor

Turbulent combustion characteristics in a one-sided diverging scramjet combustor are investigated by large eddy simulation (LES), especially for the flame regimes and turbulence/chemistry interactions. The model combustor designed by National University of Defense Technology consists of a wall-mounted cavity and an injector that allows a sonic hydrogen jet injecting into the high-enthalpy supersonic crossflow. In the simulation, an adaptive central-upwind 6th-order weighted essentially nonoscillatory (WENO-CU6) scheme along with multi-threaded and multi-process MPI/OpenMP parallel is adopted to improve the accuracy and parallel efficiency of solver. Supersonic turbulent boundary layer is generated by a multi-wall recycling/rescaling method at the inflow, in which turbulent fluctuations support a more rapid jet mixing. In addition, the chemical source term is closed by an assumed probability density function (PDF) approach that accounts for the fluctuations of temperature and composition. Validation is performed for the distribution of wall pressure and time-averaged structures, and the predicted results show in good agreement with experimental data. Through the bow shock, supersonic fluid with high enthalpy is stagnated in front of the jet plume, leading to the occurrence of auto-ignition. However, the local scalar dissipation rate here is so large that strong chemical reaction is inhibited. It is found that the combustion is anchored at the trailing edge of cavity accompanied by violent heat release, which is governed by a cavity-stabilized diffusion flame. The jet-cavity interactions extend the residence time of fluid and entrain plenty of hot intermediate products into the cavity, which is beneficial to the ignition and flame stabilization.

Large-eddy simulation of a spatially-evolving supersonic turbulent boundary layer at [formula omitted

The ability of large-eddy simulations (LES) to resolve the most energetic coherent structures of a spatially-evolving supersonic turbulent boundary layer over a flat plate at M∞=2 and Reθ≈6000 is analyzed using three different local subgrid scale models. Additionally, an Implicit LES (ILES), which relies on the intrinsic numerical dissipation to act as a subgrid model, is investigated to assess the consistency and the accuracy of the method. Direct comparison with data from high resolution DNS calculations (Pirozzoli and Bernardini, 2011) provides validation of the different modeling approaches. Turbulent statistics up to the fourth-order are reported, which help emphasizing some salient features related to near-wall asymptotic behavior, mesh resolution and models prediction. Detailed analysis of the near-wall asymptotic behavior of all relevant quantities shows that the models are able to correctly reproduce the near-wall tendencies. The thermodynamic fluctuations, Trms and ρrms, show a lack of independence from SGS modeling and grid refinement in contrast to the velocity fluctuations. The pressure fluctuations, which are associated with the acoustic mode, are not significantly affected by the modeling and the mesh resolution. Furthermore, the comparison of different contributions to the viscous dissipation reveals that the solenoidal dissipation plays the most dominant role regardless of the model. Finally, it is found that the ILES is more likely to produce consistent near-wall behavior even with a numerical scheme that has a small amount of numerical dissipation to emulate the effects of unresolved scales.

Experimental investigation on aero-optics of supersonic turbulent boundary layers

Nanoparticle-based planar laser scattering was used to measure the density distribution of the supersonic (Ma=3.0) turbulent boundary layer and the optical path difference (OPD), which is quite crucial for aero-optics study. Results were obtained using ray tracing. The influences of different layers in the boundary layer, turbulence scales, and light incident angle on aero-optics were examined, and the underlying flow physics were analyzed. The inner layer plays a dominant role, followed by the outer layer. One hundred OPDrms of the outer layer at different times satisfy the normal distribution better than that of the inner layer. Aero-optics induced by the outer layer is sensitive to the filter scale. When induced by the inner layer, it is not sensitive to the filter scale. The vortices with scales less than the Kolmogorov scale (=46.0 μm) have little influence on the aero-optics and could be ignored; the validity of the smallest optically active scale (=88.1 μm) proposed by Mani is verified, and vortices with scales less than that are ignored, resulting in a 1.62% decay of aero-optics; the filter with a width of 16-grid spacing (=182.4 μm) decreases OPDrms by 7.04%. With the increase of the angle between the wall-normal direction and the light-incident direction, the aero-optics becomes more serious, and the difference between the distribution of the OPDrms and the normal distribution increases. The difficulty of aero-optics correction is increased. Light tilted toward downstream experiences more distortions than when tilted toward upstream at the same angle relative to the wall-normal direction.

Temperature-Invariant Scaling for Compressible Turbulent Boundary Layers with Wall Heat Transfer

ABSTRACTIn a recent paper by Zhang et al. in 2012, a Mach number-invariant scaling was proposed to account for the effect of variation of free-stream Mach number in supersonic turbulent boundary layers. The present work focuses on the effect of variation of wall temperature with strong heating and cooling at the wall. Direct numerical simulation is used to study scaling and turbulence structure of a spatially evolving Mach 2 supersonic boundary layer at a friction Reynolds number of 500. A new scaling law is proposed to account for temperature-dependent fluid-property variations. This universal scaling appears superior to the existing models with the novelty that it applies not only for the mean-velocity profile but also extends to the turbulent transport, production, and dissipation terms in the budget of the turbulent kinetic energy.

Recovery of a supersonic turbulent boundary layer after an expansion corner

Supersonic turbulent flows at Mach 2.7 over expansion corners with deflection angles of 0° (flat plate), 2°, and 4° have been studied using direct numerical simulation. Distributions of skin friction, pressure, velocity, and boundary layer growth show that the turbulent boundary layer experiences a recovery from a non-equilibrium to an equilibrium state downstream of the expansion corner. Analysis of velocity profiles indicates that the streamwise velocity undergoes a reduction in the near-wall region even though the velocity in the core part of the boundary layer is accelerated after the expansion corner. Growth of the boundary layer was evaluated and a higher shape factor was found in the expansion cases. Turbulence was found to be mostly suppressed downstream of the corner, and throughout the recovery region, even though turbulence is regenerated in the near-wall region. The expansion ramp increases the near-wall streak spacing compared to a flat plate, and turbulent kinetic energy profiles and budgets exhibit a characteristic two-layer structure. Near-wall turbulence recovers to a balance between the local production and dissipation equilibrium more quickly in the inner layer than in the outer layer. The two-layer structure is due to a history effect of turbulence decay in the outer part of the boundary layer downstream of the expansion corner, with limited momentum and energy exchange between the inner layer and the main stream.

Near-wall behaviors of oblique-shock-wave/turbulent-boundary-layer interactions

A direct numerical simulation (DNS) on an oblique shock wave with an incident angle of 33.2° impinging on a Mach 2.25 supersonic turbulent boundary layer is performed. The numerical results are confirmed to be of high accuracy by comparison with the reference data. Particular efforts have been made on the investigation of the near-wall behaviors in the interaction region, where the pressure gradient is so significant that a certain separation zone emerges. It is found that, the traditional linear and logarithmic laws, which describe the mean-velocity profiles in the viscous and meso sublayers, respectively, cease to be valid in the neighborhood of the interaction region, and two new laws of the wall are proposed by elevating the pressure gradient to the leading order. The new laws are inspired by the analysis on the incompressible separation flows, while the compressibility is additionally taken into account. It is verified by the DNS results that the new laws are adequate to reproduce the mean-velocity profiles both inside and outside the interaction region. Moreover, the normalization adopted in the new laws is able to regularize the Reynolds stress into an almost universal distribution even with a salient adverse pressure gradient (APG).

Constrained large-eddy simulation of supersonic turbulent boundary layer over a compression ramp

ABSTRACTA supersonic turbulent boundary layer over a compression ramp is numerically investigated using the constrained large-eddy simulation (CLES) method. The compression corner is characterised by a deflection angle of 24°. The free-stream Mach number is Ma∞ = 2.9, and the Reynolds number based on the momentum thickness of inlet boundary layer is Reθ = 2300. The mean and statistical quantities, such as mean velocity, wall pressure and Reynolds stresses, are thoroughly analysed and compared with those from traditional large-eddy simulation (LES), experimental measurement and direct numerical simulation (DNS). It turns out that CLES can predict the friction coefficient, wall-pressure distribution, size of separation bubble, Reynolds stresses, etc. more accurately than traditional LES, and the results are in reasonable agreement with the experimental and/or DNS data. Also discussed are the effects of specific parameterisations of the Reynolds constraint and interfacial positions separating the constrained and unconstrained regions on the performance of the CLES method.

An experimental investigation of the supersonic turbulent boundary layer subjected to convex curvature

By employing particle image velocimetry, the impact of convex curvature on the supersonic turbulent boundary layer with inflow Mach number of 2.95 is experimentally investigated. The test models are carefully designed to have a zero-pressure-gradient convex boundary layer to isolate the influence of the streamwise pressure gradient while looking at the impact of convex curvature only. The results are compared with the favorable-pressure-gradient case (with pressure gradient parameter β=-0.70). For both of the zero-pressure-gradient and favorable-pressure-gradient convex boundary layers, the log law is found to be well preserved in the profile of streamwise velocity with inner scaling. Both of the mean streamwise velocity and the principal strain rate of the zero-pressure-gradient case collapse after scaling. By accumulating the streamwise velocity along the wall normal direction, the convex curvature is found to have the potential of diminishing the organized motion of the large scale vortices, which cont...

Numerical analysis of shock wave and supersonic turbulent boundary interaction between adiabatic and cold walls

ABSTRACTDirect numerical simulations of shock wave and supersonic turbulent boundary layer interaction in a 24° compression ramp with adiabatic and cold-wall temperatures are conducted. The wall temperature effects on turbulence structures and shock motions are investigated. The results are validated against previous experimental and numerical data. The effects of wall cooling on boundary layer characteristics are analysed. Statistical data show that wall cooling has a significant effect on the logarithmic region of mean velocity profile downstream the interaction region. Moreover, the influence of wall temperature on Reynolds stress anisotropy is mainly limited in the near-wall region and has little change on the outer layer. As the wall temperature decreases, the streamwise coherency of streaks increases. Based on the analysis of instantaneous Lamb vector divergence, the momentum transport between small-scale vortices on cold-wall condition is significantly enhanced. In addition, spectral analysis of wall pressure signals indicates that the location of peak of low-frequency energy shifts toward higher frequencies in cold case. Furthermore, the dynamic mode decomposition results reveal two characteristic modes, namely a low-frequency mode exhibiting the breathing motion of separation bubble and a high-frequency mode associated with the propagation of instability waves above separation bubble. The shape of dynamic modes is not sensitive to wall temperature.

Implicit large eddy simulation of acoustic loading in supersonic turbulent boundary layers

This paper investigates the accuracy of implicit large eddy simulation in the prediction of acoustic phenomena associated with pressure fluctuations within a supersonic turbulent boundary layer. We assess the accuracy of implicit large eddy simulation against direct numerical simulation and experiments for attached turbulent supersonic flow with zero-pressure gradient, and further analyze and discuss the effects of turbulent boundary layer pressure fluctuations on acoustic loading both at the high and low frequency regimes. The results of high-order variants of the simulations show good agreement with theoretical models, experiments, as well as previously published direct numerical simulations.

Numerical studies of shock wave interactions with a supersonic turbulent boundary layer in compression corner:Turning angle effects

Abstract Direct numerical simulations (DNS) were performed to investigate the interactions of a Mach 2.9 turbulent boundary layer with shock waves of varying strengths in compression corner. The supersonic turbulent boundary layer was triggered by wall blowing-and-suction perturbations. The shock waves were produced by two-dimensional compression corners of 8, 14, 20 and 24°. Compared with previous DNS results and experimental data, the numerical calculations were validated. The effects of shock wave on the boundary layer are studied by both flow visualizations and statistical analysis, and the results show that the intensity of fluctuations is amplified greatly by the shock wave. With the increasing of turning angle, three-dimensionality of separation bubble is significantly enhanced. Based on the statistics and power spectrum of the wall pressure signals, the effect of turning angle on the unsteadiness of shock motion is also studied, and the results show that the shock motions are quite different in the small and the large turning angle cases. The motion in the 8° and 14° cases is characterized by high-frequency and small-amplitude, but the low-frequency and large-scale streamwise oscillation is the main feature in the 20° and 24° cases. The effect of turning angle on the turbulence state is analyzed by using the anisotropy of Reynolds stress tensor. The coherent vortex structures are also studied qualitatively. The results indicate that the cane-like streamwise vortexes in the near-wall region are the dominant structure for the small angle cases, while the hairpin vortexes and packets in the outer layer play the leading role in the large angle cases. According to the quantitative analysis of turbulent kinetic energy budgets in the separation region, the effect of turning angle on the transport mechanism is studied. It is found that the influence of shear layer above separation bubble on the mechanism is significant.

Image pre-processing method for near-wall PIV measurements over moving curved interfaces

PIV measurements near a moving interface are always difficult. This paper presents a PIV image pre-processing method that returns high spatial resolution velocity profiles near the interface. Instead of re-shaping or re-orientating the interrogation windows, interface tracking and an image transformation are used to stretch the particle image strips near a curved interface into rectangles. Then the adaptive structured interrogation windows can be arranged at specified distances from the interface. Synthetic particles are also added into the solid region to minimize interfacial effects and to restrict particles on both sides of the interface. Since a high spatial resolution is only required in high velocity gradient region, adaptive meshing and stretching of the image strips in the normal direction is used to improve the cross-correlation signal-to-noise ratio (SN) by reducing the velocity difference and the particle image distortion within the interrogation window. A two dimensional Gaussian fit is used to compensate for the effects of stretching particle images. The working hypothesis is that fluid motion near the interface is ‘quasi-tangential flow’, which is reasonable in most fluid-structure interaction scenarios. The method was validated against the window deformation iterative multi-grid scheme (WIDIM) using synthetic image pairs with different velocity profiles. The method was tested for boundary layer measurements of a supersonic turbulent boundary layer on a flat plate, near a rotating blade and near a flexible flapping flag. This image pre-processing method provides higher spatial resolution than conventional WIDIM and good robustness for measuring velocity profiles near moving interfaces.

Experimental investigation on aero-optics of supersonic turbulent boundary layers at different light incident angles

The aero-optical distortion caused by the compressibility of high-speed flow field has a great influence on the development of airborne optical detection system of (hypersonic) supersonic vehicles. The turbulent boundary layer is one of the most important aspects in the aero-optical study, and has become one of the hot research points in the field of aero-optical study. The nano-particle-based planar laser scattering technique is used to measure the density distribution of the supersonic (Ma=3.0) turbulent boundary layers, and the optical path difference, which is quite crucial for the aero-optical study, is obtained by ray-tracing method. The experimental result is verified by being compared with the theoretical result computed by the aero-optical scaling method of turbulent boundary layers. Five different light incident angles (α=60°, 75°, 90°, 105°, 120°) are selected and used to examine the influences of light incident angles on the supersonic turbulent layer, and the underlying flow physics is analyzed. Research shows that the light propagation path in the supersonic turbulent boundary layer contributes to the light incident angle dependence of aero-optics. The different propagation paths lead to the difference between the light propagation distance in the flow field and the correlation results of the corresponding density fluctuation. The oblique incidence of light results in the increase of the propagation distance in the flow field, and then the aero-optics turns worse. The greater the angle between the incident direction of light and the vertical direction of the wall, the more significant the aero-optics is, the difference increases at different times, the difficulty in correcting the aero-optics is also increased. In the supersonic turbulent boundary layer, a large number of vortices with a specific orientation lead to the anisotropy of the aero-optics in the turbulent boundary layer. By calculating the spatial two-point correlation of the density fluctuations at the streamwise plane (x-y plane), the cross-correlation result of density fluctuations at any light incidence angle (α=0°-180°) can be obtained. The local coherent structure scale is nearly 0.20 mm, which is basically consistent with the aero-optical effective scale (≈ 0.18 mm) computed from the formula proposed by Mani et al. When the light is inclined downstream, the direction of light propagation is consistent with the vortex structure in the flow field, and in this direction, the correlation coefficient of density fluctuation is larger, so the aero-optics is more serious. When the light beam is tilted upstream, the correlation coefficient is smaller, so the aero-optics is weaker.

Experimental study on the density characteristics of a supersonic turbulent boundary layer

An experimental study on the density characteristics of a zero-pressure-gradient flat plate turbulent boundary layer at Ma=3.0 is performed by the density field measurement method based on Nano-tracer planar laser scattering (NPLS) technology. The mean and the fluctuating characteristics of the density field of the boundary layer are analyzed. And the spectrum analyses of density fluctuations are performed by utilizing Taylor's hypothesis to convert spatial measurements into pseudo-temporal measurements. The mean density profile increases away from the wall, which accords well with the density profile deduced from the mean velocity distribution by using the adiabatic Crocco-Busemann relation. The root mean square (RMS) of the density fluctuations increases in the logarithmic region with a peak value of 0.2ρ∞, and its probability density distribution follows a normal distribution. However, the RMS of density fluctuations decreases in the outer region of the boundary layer. According to the spectrum analysis, the density fluctuations are characterized in a wide range of frequencies throughout the boundary layer, with the maximum frequency on the order of 1 MHz. The low frequency fluctuations are predominant near the wall and in the outer region of the turbulent boundary layer. However, the proportion of high-frequency fluctuations is nearly equal to that of low-frequency fluctuations in the logarithmic region. The combined NPLS and PIV technique provide a simultaneous density and velocity measurements of the present turbulent boundary layer. The high frequency fluctuations in the supersonic turbulent boundary layer may be induced by the density fluctuations, which are caused by the convection of the turbulent structures with nonuniform density distributions. And the contribution of the velocity fluctuations only to the low frequency fluctuations is observed. There are good similarities between the density fluctuations and the mass flux fluctuations for both the probability density distribution and the spectrum characteristics. On the contrary, a large difference between the fluctuations of velocity and density is identified. Therefore, the strong density fluctuations inside supersonic turbulent boundary layers, as well as its difference between the velocity fluctuations, should be one of the most important differences between compressible and incompressible turbulent boundary layers.

Structural responses of the supersonic turbulent boundary layer to expansions

Structural responses of the supersonic turbulent boundary layer to the expansions induced by a convex wall and a ramp are experimentally investigated. Relaminarization of part of the turbulent boundary layer in the near wall region is clearly visualized, which has been seldom presented before. The relaminarized layers formed over two test models are different. While a thicker relaminarized layer is observed for the ramp, a longer lasting layer is noticed for the convex wall. The structure angle is found to be increased by the expansions. Increases of turbulence scale and boundary layer thickness are observed. The contribution of the bulk dilatation to the boundary layer growth is stronger than that of the centrifugal force.

An experimental investigation of the supersonic turbulent boundary layer subjected to concave curvature

By employing particle image velocimetry, the response of a Mach 2.95 turbulent boundary layer to the concave curvature is experimentally investigated. The radius of the concave wall is 350 mm, and the turning angle is 20∘. Logarithmic law is well preserved in the profile of streamwise velocity at all streamwise positions despite the impact of curvature. The varying trend of principal strain rate is found to be different at different heights within the boundary layer, which cannot be explained by the suggestion given by former researchers. Based on the three-layer model proposed in this paper, distribution of the principal strain rate is carefully analyzed. The streamwise increase of wall friction is suggested to be brought by the increase of velocity gradient in the thin subsonic layer. Increases of the static temperature and the related sound speed are responsible for that. Larger correlated turbulent motions could be introduced by the concave curvature. The probability density histograms of streamwise velocity reveal that the large scale hairpin packets are statistically well organized. The concave curvature is found to have the potential of reinforcing the organization, which explains the increase of turbulent level in the supersonic concave boundary layer.

Assessment of subgrid-scale modeling for large-eddy simulation of a spatially-evolving compressible turbulent boundary layer

Abstract The performance of three standard subgrid-scale (SGS) models, namely the wall-adapting local eddy-viscosity (WALE) model, the Dynamic Smagorinsky model (DSM) and the Coherent Structures model (CSM), are investigated in the case of a spatially-evolving supersonic turbulent boundary layer (STBL) over a flat plate at M ∞ = 2 and Reθ ≈ 2600. A high-order split-centered scheme is used to discretize the convective fluxes of the Navier–Stokes equations, and is found to be highly effective to overcome the dissipative character of the standard shock-capturing WENO scheme. The consistency and the accuracy of the simulations are evaluated using direct numerical simulations taken from the literature. It is demonstrated that all SGS models require a comparable minimum grid refinement in order to capture accurately the near-wall turbulence. Overall, the models exhibit correct behavior when predicting the dynamic properties, but show different performances for the temperature distribution in the near-wall region even for cases with satisfactory energy resolution of more than 80%. For a well-resolved LES, the SGS dissipation due to the fluctuating velocity gradients is found to dominate the total SGS dissipation.