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The interaction length induced by Shock Wave/Turbulent Boundary-Layer Interactions (SWTBLIs) in the hypersonic flow was investigated using a scaling analysis, in which the interaction length normalized by the displacement thickness of boundary layer was correlated with a corrected non-dimensional separation criterion across the interaction after accounting for the wall temperature effects. A large number of hypersonic SWTBLIs were compiled to examine the scaling analysis over a wide range of Mach numbers, Reynolds numbers, and wall temperatures. The results indicate that the hypersonic SWTBLIs with low Reynolds numbers collapse on the supersonic SWTBLIs, while the hypersonic cases with high Reynolds numbers show a more rapid growth of the interaction length than that with low Reynolds numbers. Thus, two scaling relationships are identified according to different Reynolds numbers for the hypersonic SWTBLIs. The scaling analysis provides valuable guidelines for engineering prediction of the interaction length, and thus, enriches the knowledge of hypersonic SWTBLIs.

In order to understand the physical phenomenon of the reflected shock/turbulent boundary layer interaction, the Large Eddy Simulation (LES) is conducted to investigate shock wave and turbulent boundary layer interaction in a 12 ° compression ramp with inlet high Mach number of 2.9. Rescaling/recycling method is used as inflow turbulence generation technique and validated on a supersonic flat plate turbulent boundary layer. The flow field of recycling plane in the plate computation domain is obtained to give the inlet boundary condition for the LES computation. This paper focuses on the reflected shock/turbulent boundary layer interaction region, where the fine flow structure and instantaneous flow field are analyzed in detail. It is found that the unsteady motion of the shock wave leads to the increase of wall pressure fluctuation.

International audience; In order to be able to judge the effectiveness of transition induction in WP-2, reference flow cases were planned in WP-1. There are two obvious reference cases—a fully laminar interaction and a fully turbulent interaction. Here it should be explained that the terms “laminar” and “turbulent” interaction refer to the boundary layer state at the beginning of interaction only. There are two basic configurations of shock wave boundary layer interaction and these are a part of the TFAST project. One is the normal shock wave, which typically appears at the transonic wing and on the turbine cascade. The characteristic incipient separation Mach number range is about M = 1.2 in the case of a laminar boundary layer and about M = 1.32 in the case of turbulent boundary layer. The second typical flow case is the oblique shock wave reflection. The most characteristic case in European research is connected to the 6th FP IP HISAC project concerning a supersonic business jet. The design speed of this airplane is M = 1.6. Therefore the TFAST consortium decided to use this Mach number as the basic case. Pressure disturbance at this Mach number is not very high and can be compared to the disturbance of the normal shock at the incipient separation Mach number mentioned earlier. As mentioned earlier, shock reflection at M = 1.6 may be related to incipient separation. Therefore two additional test cases were planned with different Mach numbers. ITAM conducted an M = 1.5 test case, and TUD an M = 1.7 test case. These partners have also previously made very specialized and successful contributions to the UFAST project.

Upon its development and initial characterization, the supersonic variant of the nanoscale thermal anemometry probe (S-NSTAP) was deployed in a supersonic wind tunnel facility, where both freestream and boundary layer measurements were obtained at $$M_\infty =2$$ . The low operating stagnation pressures generated reliable data, where the effects of Reynolds number, Mach number and overheat ratio on the sensor’s heat transfer were investigated in detail. The performance of the S-NSTAP was also compared to that of a conventional cylindrical hot-wire and the S-NSTAP was shown to exhibit unparalleled temporal resolution ( $$\sim $$ 300 kHz). The mass flux sensitivity coefficient of both hot-wires was further computed and appeared to vary between probes, yielding a coefficient twice as large for the conventional probe than for the S-NSTAP. The experimental data obtained from both hot-wires were also compared, via spectral analysis and turbulence statistics, to the results of a numerically modelled turbulent boundary layer.

The paper presents the experimental results and the results of direct numerical simulation of the development and interaction of two wave trains from two point sources of controlled disturbances in a supersonic boundary layer on a flat plate with an incident flow Mach number of 2.5. Sources were located parallel to the leading edge of the model. For the introduction of controlled disturbances into the boundary layer, the normal component of the mass flow rate was varied in the calculations. In the experiment, periodic glow discharges at a frequency of 20 kHz were used. In both cases, the disturbances sources worked synchronously. Mass flow rate pulsations were measured and recorded in sections, parallel to the leading edge of the model, near the maximum of disturbances along the boundary layer. In the experiment, a constant-temperature hot-wire anemometer was used. After performing a discrete Fourier transform, the spatial distributions of disturbances, the beta-spectra were determined, and the wave characteristics of the development of disturbances downstream were estimated. In addition, direct numerical simulation of the downstream development of disturbances from a single source was performed. The work presents a comparison of experimental and theoretical calculated data. The paper discusses the effects inherent in the interaction of unstable traveling controlled disturbances from two sources operating synchronously.

This work attempts to connect internal flow to the exit flow and supersonic jet mixing in rectangular nozzles with low to high aspect ratios (AR). A series of low and high aspect ratio rectangular nozzles (design Mach number = 1.5) with sharp throats are numerically investigated using steady state Reynolds-averaged Navier−Stokes (RANS) computational fluid dynamics (CFD) with k-omega shear stress transport (SST) turbulence model. The numerical shadowgraph reveals stronger shocks at low ARs which become weaker with increasing AR due to less flow turning at the throat. Stronger shocks cause more aggressive gradients in the boundary layer resulting in higher wall shear stresses at the throat for low ARs. The boundary layer becomes thick at low ARs creating more aerodynamic blockage. The boundary layer exiting the nozzle transforms into a shear layer and grows thicker in the high AR nozzle with a smaller potential core length. The variation in the boundary layer growth on the minor and major axis is explained and its growth downstream the throat has a significant role in nozzle exit flow characteristics. The loss mechanism throughout the flow is shown as the entropy generated due to viscous dissipation and accounts for supersonic jet mixing. Axis switching phenomenon is also addressed by analyzing the streamwise vorticity fields at various locations downstream from the nozzle exit.

Microramp vortex generators are robust, reliable, and simple devices for passively controlling the boundary layer in several aerospace applications. Various past studies have investigated the effectiveness of microramps in controlling the flow separation induced by shock-wave/boundary-layer interactions. Building upon the past knowledge, this paper reports a systematic investigation of the relation between the microramp geometry and the downstream flow characteristics. A simplified flow model of the microramp wake is provided to explain and predict the influence of changing the geometry on the circulation of the primary vortex pair. The model also provides scaling relations for the evolution of the wake characteristics (that is, wake velocity, wake location, and added incompressible momentum), incorporating the effect of all geometry parameters. Extensive experimental data have been used to validate the model.

A study of dense-gas effects on the laminar, transitional and turbulent characteristics of boundary layer flows is conducted. The laminar similarity solution shows that temperature variations are small due to the high specific heats of dense gases, leading to velocity profiles close to the incompressible ones. Nevertheless, the complex thermodynamics of the base flow has a major impact on unstable modes, which bear similarities with those obtained for a strongly cooled wall. Numerical simulations of spatially developing boundary layers yield turbulent statistics for the dense gas flow that remain closer to the incompressible regime than perfect gas ones despite the presence of strongly compressible structures.

Abstract Shock-wave/boundary-layer interactions (SW–BLIs) play an important role in a wide range of transonic, supersonic and hypersonic applications. Fundamental studies on stationary interactions have been conducted extensively during the last 60 years. However, unsteady SWBLIs with traveling shock fronts have been little studied on canonical geometries. In the present experimental investigation, the influence of a uniformly moving impinging shock on the separated SWBLI flow is analyzed, with a freestream Mach number of 3 and a traveling Mach number in upstream direction of 0.5. To evaluate this effect, stationary reference SWBLIs have been investigated in a wide ranging study. A scaling method from the literature has been enhanced to drastically reduce the data scattering using a new approach accounting for the Reynolds number influence. The results gathered from the traveling interactions were within the spread of reference data, considering the true shock-wave Mach number of 3.5. The validity of the modified scaling approach to describe the interaction length in cases with steady and traveling shock-wave/turbulent-boundary-layer interactions is discussed. Graphic abstract

In order to achieve proficient combustion with the present technologies, the flow through an aircraft intake operating at supersonic and hypersonic Mach numbers must be decelerated to a low-subsonic level before entering the combustion chamber. High-speed intakes are generally designed to act as a flow compressor even in the absence of mechanical compressors. The reduction in flow velocity is essentially achieved by generating a series of oblique as well as normal shock waves in the external ramp region and also in the internal isolator region of the intake. Thus, these intakes are also referred to as mixed-compression intakes. Nevertheless, the benefits of shock-generated compression do not arise independently but with enormous losses because of the shockwave and boundary layer interactions (SBLIs). These interactions should be manipulated to minimize or alleviate the losses. In the present investigation a wall ventilation using a new cavity configuration (having a cross-section similar to a truncated rectangle with the top wall covered by a thin perforated surface is deployed underneath the cowl-shock impinging point of the Mach 2.2 mixed-compression intake. The intake is tested for four different contraction ratios of 1.16, 1.19, 1.22, and 1.25, with emphasis on the effect of porosity, which is varied at 10.6%, 15.7%, 18.8%, and 22.5%. The introduction of porosity on the surface covering the cavity has been proved to be beneficial in decreasing the wall static pressure substantially as compared to the plain intake. A maximum of approximately 24.2% in the reduction in pressure at the upstream proximal location of 0.48 L is achieved in the case of the wall-ventilated intake with 18.8% porosity, at the contraction ratio of 1.19. The Schlieren density field images confirm the efficacy of the 18.8% ventilation in stretching the shock trains and in decreasing the separation length. At the contraction ratios of 1.19, 1.22, and 1.25 (&lsquo, dual-mode&rsquo, contraction ratios), the controlled intakes with higher porosity reduce the pressure gradients across the shockwaves and thereby yields an &lsquo, intake-start&rsquo, condition. However, for the uncontrolled intake, the &lsquo, unstart&rsquo, condition emerges due to the formation of a normal shock at the cowl lip. Additionally, the cowl shock in the &lsquo, intake is shifted upstream because of higher downstream pressure.

The resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) is applied to supersonic turbulent boundary layers to study the validity of Morkovin’s hypothesis, which postulates that high-speed turbulence structures in zero-pressure-gradient turbulent boundary layers remain largely the same as their incompressible counterparts. Supersonic zero-pressure-gradient turbulent boundary layers with adiabatic wall boundary conditions at Mach numbers ranging from 2 to 4 are considered. Resolvent analysis highlights two distinct regions of the supersonic turbulent boundary layer in the wave parameter space: the relatively supersonic region and the relatively subsonic region. In the relatively supersonic region, where the flow is supersonic relative to the free-stream, resolvent modes display structures consistent with Mach wave radiation that are absent in the incompressible regime. In the relatively subsonic region, we show that the low-rank approximation of the resolvent operator is an effective approximation of the full system and that the response modes predicted by the model exhibit universal and geometrically self-similar behaviour via a transformation given by the semi-local scaling. Moreover, with the semi-local scaling, we show that the resolvent modes follow the same scaling law as their incompressible counterparts in this region, which has implications for modelling and the prediction of turbulent high-speed wall-bounded flows. We also show that the thermodynamic variables exhibit similar mode shapes to the streamwise velocity modes, supporting the strong Reynolds analogy. Finally, we demonstrate that the principal resolvent modes can be used to capture the energy distribution between momentum and thermodynamic fluctuations.

Boundary layer transition is a hot research topic in fluid mechanics and aerospace engineering. In low-speed flows, two-dimensional Tollmien-Schlichting (T-S) waves always dominate the flow instability, which has been modeled by Coder and Maughmer from 2013. However, in supersonic flows, three-dimensional oblique Tollmien-Schlichting waves become dominant in flow instability. Inspired by Coder and Maughmer’s NTS amplification factor transport equation for two-dimensional Tollmien-Schlichting waves in low-speed flows and Kroo and Sturdza’s linear stability theory (LST) analysis results for oblique Tollmien-Schlichting waves in supersonic flows, a new amplification factor transport equation for oblique Tollmien-Schlichting waves has been developed based on LST. The compressible Falkner-Skan similarity equations are introduced to build the relationships between nonlocal variables and local variables so that all the variables used in the present model can be calculated using local variables. Applications of this new transport equation to the flows over supersonic flat plate, 3% thick biconvex airfoil, and one modified supersonic laminar airfoil show promising results compared with the standard LST analysis results.

Supersonic inlets are a key component of present and future air-breathing propulsion systems for high-speed flight. The inlet design is challenging because of several phenomena that must be taken under control: shock waves, boundary layer separation and unsteadiness. Furthermore, the intensity of these phenomena is strongly influenced by the working conditions and so active control systems can be particularly useful in off-design conditions. In this work, a mixed compression supersonic inlet with a double wedge ramp is considered. The flow field was numerically investigated at different values of Mach number. The simulations show that large separations appear at the higher Mach numbers on both the upper and lower walls of the duct. In order to improve the performances of the inlet two different control strategies were investigated: plasma actuators and bleed. Different locations of the plasma actuator are considered in order to also apply this technology to configurations with a diverter which prevents boundary layer ingestion. The potential of the proposed solutions is investigated in terms of total pressure recovery, flow distortion and power consumption.

The development of primary and secondary instabilities is investigated numerically for a supersonic transitional flow over a backward-facing step at Ma = 1.7 and Reδ0 = 13 718. Oblique Tollmien-Schlichting (T-S) waves with properties according to linear stability theory (LST) are introduced at the domain inlet with zero, low, or high amplitude (cases ZA, LA, and HA). A well-resolved large eddy simulation (LES) is carried out for the three cases to characterize the transition process from laminar to turbulent flow. The results for the HA case show a rapid transition due to the high initial disturbance level such that the non-linear interactions already occur upstream of the step, before the Kelvin-Helmholtz (K-H) instability could get involved. In contrast, cases ZA and LA share a similar transition road map in which transition occurs in the separated shear flow behind the step. Case LA is analyzed in detail based on the results from LST and LES to scrutinize the evolution of T-S, K-H, and secondary instabilities, as well as their interactions. Upstream of the step, the linear growth of the oblique T-S waves is the prevailing instability. Both T-S and K-H modes act as the primary mode within a short distance behind the step and undergo a quasilinear growth with a weak coupling. Upon pairing of the large K-H vortices, subharmonic waves are produced, and secondary instabilities begin to dominate the transition. Simultaneously, the growth of T-S waves is retarded by the slow resonance between subharmonic K-H and secondary instabilities. The vortex breakdown and reattachment downstream further contribute to the development of the turbulent boundary layer.

In this paper comparative flow field analysis of two intake configuration i.e. Boundary Layer Diverter Intake and Diverterless Supersonic Intake is carried out based on dimensionless parameters under various flow conditions. Numerical analysis of aircraft intake is a complex phenomenon which involves both external and internal flow analysis. In this research, both external and internal flow characteristics of intake duct are analyzed in detail. A comprehensive mesh scheme is devised and implemented to accurately capture the flow behavior in external surrounding of intake duct and flow passing through the intake duct. The analysis is carried out at different flow conditions to analyze the flow behavior in subsonic and supersonic regimes. Engine design mass flow rate is used for accurate intake analysis and results are validated with available literature. Boundary layer diversion and pressure recovery are examined for each intake configuration and comparative analysis based on pressure recovery is carried out subsequently. The analysis reveals that at subsonic and transonic regimes, Boundary Layer Diverter intake is much more effective than Diverter less Supersonic Intake, however, in supersonic regime Diverter less Supersonic Intake is found be to more effective. The research can further help in modifying/ improving the design of an existing intake configuration for enhanced intake efficiency.

The effectiveness of streak modes in controlling the oblique-type breakdown in a supersonic boundary-layer at Mach 2.0 is investigated using direct numerical simulations. Investigations in the literature have shown the effectiveness of streak modes in delaying the onset of transition dominated by two-dimensional waves, but in oblique breakdown, three-dimensional waves and a strong streak mode dominate the transition process. Paredes et al. (J. Fluid Mech., vol. 831, 2017, pp. 524–553) discussed the possible stabilization of supersonic boundary layers by optimally growing streaks using parabolized stability equations. However, no study has as yet been reported regarding direct nonlinear control of oblique breakdown. This study deals with the effects of large-amplitude decaying streak modes generated by a blowing–suction strip at the wall to control full breakdown in a reference case. Modes with four to five times the fundamental wavenumber are found to be beneficial for controlling the transition. In the first region after the control-mode forcing, the beneficial mean-flow distortion (MFD), generated by inducing the control mode, is solely responsible for hampering the growth of the fundamental-mode. On the whole, the MFD and the three-dimensional part of the control contribute equally towards controlling the oblique breakdown. The results show significant suppression of transition, and substantial improvements have been observed in the levels of the skin-friction coefficient and wall-temperature in comparison to the uncontrolled case. Moreover, refreshing the control using an additional downstream control strip increases the gain. However, the forcing amplitude must be carefully chosen in order not to introduce a generalized inflection point in the spanwise averaged mean flow invoking enhanced disturbance growth.

Direct numerical simulations (DNS) are performed for the supersonic boundary layers (SBLs) with a free-stream Mach number M ∞ = 2.2 . Different cases including the adiabatic and the isothermal (cooled and heated) walls are investigated. The laminar boundary layer is excited by means of a blowing and suction strip with single-frequency and multiple spanwise wave-numbers. The incoming laminar flow is strongly perturbed with a perturbation intensity of 2.4% of the free-stream velocity to obtain the turbulent boundary layer. In the fully developed turbulent regions, the joint probability density function (JPDF) distribution and the covariance integrands’ analyses of different parameters are performed to find out the contribution of various physical mechanisms towards different transfer processes. The results reveal that behavior of the turbulent shear stress is similar to its incompressible counterpart and the wall-temperature impacts are dominant in the buffer layer region (at y + = 10). The inclination angles of coherent structures show variations arising from the wall-temperature in both the buffer-layer and the log region. The covariance integrands’ analyses of different components of the heat flux reveal the dominance of a different transfer process in case of the cooled wall, and as a result of this difference, the cooled wall acts as a heat sink.

This paper presents our recent work on investigating velocity slip boundary conditions&rsquo, effects on supersonic flat plate boundary layer flow stability. The velocity-slip boundary conditions are adopted and the flow properties are obtained by solving boundary layer equations. Stability analysis of two such boundary layer flows is performed by using the Linear stability theory. A global method is first utilized to obtain approximate discrete mode values. A local method is then utilized to refine these mode values. All the modes in these two scenarios have been tracked upstream-wisely towards the leading edge and also downstream-wisely. The mode values for the no-slip flows agree well with the corresponding past results in the literature. For flows with slip boundary conditions, a stable and an unstable modes are detected. Mode tracking work is performed and the results illustrate that the resonance phenomenon between the stable and unstable modes is delayed with slip boundary conditions. The enforcement of the slip boundary conditions also shortens the unstable mode region. As to the conventional second mode, flows with slip boundary conditions can be more stable streamwisely when compared with the results for corresponding nonslip flows.

International audience; In this paper, the origin and the properties of the oscillation modes in screeching non-ideally expanded rectangular jets are investigated using compressible implicit LES of rectangular supersonic jets. At the exit of a converging diverging rectangular nozzle of aspect ratio 2 and of design Mach number 1.5, the jets are under- and over-expanded. Seven simulations with four different temperature ratios ranging from 1 to 3 and two different nozzle pressure ratios are performed. The geometry of the nozzle and the exit conditions are chosen such that to match the experimental study conducted at the University of Cincinnati. First, the over-expanded jets are studied. It is shown that the total number of shock cells decreases with increased temperature ratio. However, the temperature does not influence the size of the first shock cell and the linear decrease of the shock cell size in the downstream direction. The spreading of the jet is observed to be higher along the minor axis plane than along the major axis plane. The intensity of the screech noise increases with the temperature ratio in the present study although the opposite is observed in the experiments. Moreover, for jet temperature ratios of 2.5 and 3, the strong flapping motion of the jet along the minor axis plane due to the screech feedback mechanism yields to an antisymmetric organization of the Mach wave radiation. Thereafter, the near- and far-field acoustic are studied. In the near-field, screech tones are captured, whose frequencies are consistent with both experimental data and theoretical models. In the far-field, four acoustic components typical of non-ideally expanded supersonic jets are observed, namely the screech noise, the broadband shock-associated noise, the mixing noise and the Mach wave noise. Their directivities and frequencies are in agreement with experimental results and models. The mechanism of the screech noise generation is studied by using a Fourier decomposition of the pressure field. For the four over-expanded jets, a flapping motion along the diagonal or along the minor axis plane of the jet is noted. Finally, the hypothesis that the acoustic waves completing the feedback loop in these jets are linked to the upstream-propagating acoustic wave modes of the equivalent ideally expanded jets is tested. Using a jet vortex sheet model to describe the dispersion relations of these modes, it is found that this hypothesis allows us to explain the antisymmetric jet oscillation observed at the screech frequencies. Based on frequency-wavenumber decomposition of the pressure fluctuations in the jets, it is shown that at the screech frequencies, acoustic waves propagating in the upstream direction at the ambient speed of sound exist also in the jet flow, additionally to the acoustic waves propagating outside of the jet. These acoustic waves belong to the neutral acoustic wave modes of the equivalent ideally expanded jet. These results support the idea that a vortex sheet model of the corresponding 2-D planar ideally expanded jet is capable of predicting the wave modes of a non-ideally expanded rectangular supersonic jet. They also suggest that these waves are involved in the feedback part of the screech mechanism; explaining why, for the simulated screeching rectangular jets, the associated oscillation mode is antisymmetric.

International audience; In this study, the methodology relying on LES for the generation of the acoustic sources coupled with an Euler solver for their propagation, previously developed for a free hot supersonic jet, has been applied to a similar jet but impinging on a plate with an exhaust hole. Attention has been paid to the turbulent state of the nozzle exit boundary layer, which is a key parameter for jet noise simulation.

International audience; The flutter corresponds to an aerodynamic loading of the structure which amplifies the natural blade vibration. In this paper, a modern design of a high pressure compressor is investigated using a time-linearized RANS solver on 2D blade to blade channel. Two operating points at part speed have been selected, the first with only small supersonic pockets and the second with the interblade channel blocked. Two vibration modes are investigated, the first torsion mode (with a nodal diameter at 2) and the first flexion mode (with a nodal diameter at 2, 4 and 6). Two different two equations turbulence models, k-l and k-ω have been used to resolve the steady state. The unsteady resolution is based on the previous steady state field. Turbulent variables are calculated over time based on a k-ω turbulence model. It was found that for some mode shapes, but not for all, the work exchange between the flow and the blade presents a large disparity depending on the turbulence model used primarily in the steady calculation. This paper proposes a parametric study in terms of rotor velocities, nodal diameters and vibration mode shapes to determine which flow phenomena are sensitive to the turbulence modelling. Main results point to the effect of the shockwave motion, and its interaction with the boundary layer and its separation.

The experimental study of the effect of weak shock waves on the supersonic boundary layer of the flat plate with a blunt leading edge (the radius of bluntness was r = 2.5 mm) with Mach number M = 2.5 and zero angle of attack was carried out. The measurements were carried out using the constant temperature anemometer. The paper presents a complex flow structure on the surface of the model. High-intensity peaks were found in the regions of the disturbed flow. Also the spectral analysis of perturbations was performed. It is found that the supersonic boundary layer on a flat plate is very sensitive to the effect of weak shock waves.

International audience; The transonic flow field around a supercritical airfoil is investigated. The objective of the present paper is to enhance the understanding of the physical mechanics behind two-dimensional transonic buffet. The paper is composed of two parts. In the first part, a global stability analysis based on the linearized Reynolds-averaged Navier–Stokes equations is performed. A recently developed technique, based on the direct and adjoint unstable global modes, is used to compute the local contribution of the flow to the growth rate and angular frequency of the unstable global mode. The results allow us to identify which zones are directly responsible for the existence of the instability. The technique is firstly used for the vortex-shedding cylinder mode, as a validating case. In the second part, in order to confirm the results of the first part, a selective frequency damping method is locally used in some regions of the flow field. This method consists of applying a low-pass filter on selected zones of the computational domain in order to damp the fluctuations. It allows us to identify which zones are necessary for the persistence of the instability. The two different approaches give the same results: the shock foot is identified as the core of the instability; the shock and the boundary layer downstream of the shock are also necessary zones while damping the fluctuations on the lower surface of the airfoil; and outside the boundary layer between the shock and the trailing edge or above the supersonic zone does not suppress the shock oscillation. A discussion on the several physical models, proposed until now for the buffet phenomenon, and a new model are finally offered in the last section.

3D optimal forcing and response of a 2D supersonic boundary layer are obtained by computing the largest singular value and the associated singular vectors of the global resolvent matrix. This approach allows to take into account both convective-type and component-type non-normalities responsible for the non-modal growth of perturbations in noise selective amplifier flows. It is moreover a fully non-parallel approach that does not require any particular assumptions on the baseflow. The numerical method is based on the explicit calculation of the Jacobian matrix proposed by Mettot et al. [1] for 2D perturbations. This strategy uses the numerical residual of the compressible Navier-Stokes equations imported from a finite-volume solver that is then linearised employing a finite difference method. Extension to 3D perturbations, which are expanded into modes of wave number, is here proposed by decomposing the Jacobian matrix according to the direction of the derivatives contained in its coefficients. Validation is performed on a Blasius boundary layer and a supersonic boundary layer, in comparison respectively to global and local results. Application of the method to a boundary layer at M = 4.5 recovers three regions of receptivity in the frequency-transverse wave number space. Finally, the energy growth of each optimal response is studied and discussed.

Studies have shown that micro-wedge vortex generators (MVG) can effectively control the flow separation of supersonic boundary layer. In order to improve the flight stability of spinning projectile, the original standard 155 mm projectile was taken as an example, and the micro-vanes were mounted at the projectile shoulder to investigate the separation control on the aerodynamic characteristics of projectile. Numerical simulations were performed with the use of DES method for the flow fields of projectiles with and without micro-vanes, and the characteristics of the boundary layer structures and aerodynamic data were compared and discussed. Numerical results show that the micro-vanes can be used to inhibit separation of fluid on projectile surface, and improve the flight stability and firing dispersion of projectile.

In the present work the propagating modes of detonation wave in supersonic hydrogen–air mixtures are investigated in narrow rectangular channels. To clarify the effect of the detonation wave interaction with the boundary layer on the evolution and propagation of detonation phenomenon, high-speed laser schlieren experiments and adaptive Navier–Stokes (NS) simulations (pseudo-DNS) combined with a detailed reaction model are performed. The experimental results show that after successful ignition, two propagating modes are observed and can be classified as Oblique shock-induced combustion/Mach stem-induced detonation (OSIC/MSID) and pure Oblique shock-induced combustion (OSIC). For the OSIC/MSID mode, a Mach stem induced overdriven detonation is generated in the middle of the main flow. For the pure OSIC mode, no detonation wave but two oblique shock-induced combustion regions are generated throughout the whole channel with the overall structure taking a thwartwise V shape. The OSIC/MSID and pure OSIC propagation modes are further confirmed by pseudo-DNS employing a detailed reaction model and dynamic adaptive mesh refinement for the same conditions as utilized in the experiments. The numerical results show that because of subsonic combustion near the walls induced by the boundary layers, the OSIC/MSID is not entirely symmetrical, while for the pure OSIC mode, larger fluctuations are observed along the oblique shock waves resulting from enhanced instabilities due to additional chemical heat release.

The aim of this work is to highlight the significance of Fluid–Thermal–Structural Interaction (FTSI) as a diagnosis of existing designs, and as a means of preliminary investigation to ensure the feasibility of new designs before conducting experimental and field tests. The novelty of this work lies in the multi-physics simulations, which are, for the first time, performed on rectangular nozzles. An existing experimental supersonic rectangular converging/diverging nozzle geometry is considered for multi-physics 3D simulations. A design that has been improved by eliminating the sharp throat is further investigated to evaluate its structural integrity at design Nozzle Pressure Ratio (NPR 3.67) and off-design (NPR 4.5) conditions. Static structural analysis is performed by unidirectional coupling of pressure loads from steady 3D Computational Fluid Dynamics (CFD) and thermal loads from steady thermal conduction simulations, such that the simulations represent the experimental set up. Structural deformation in the existing design is far less than the boundary layer thickness, because the impact of Shock wave Boundary Layer Interaction (SBLI) is not as severe. FTSI demonstrates that the discharge coefficient of the improved design is 0.99, and its structural integrity remains intact at off-design conditions. This proves the feasibility of the improved design. Although FTSI influence is shown for a nozzle, the approach can be applied to any product design cycle, or as a prelude to building prototypes.

An experimental investigation has been conducted to study the unsteady nature of the shock structure in an over-expanded planar nozzle. The main emphasis was to investigate (i) the effect of resonance associated with laminar shock-wave/boundary-layer interactions on the unsteady characteristics of the wall pressure signals in the region of the interaction and, (ii) the change in these characteristics as the state of boundary layer undergoes transition to turbulence and a resonance free interaction. The tests were conducted in a Mach 2 planar nozzle with wall-half angle of 5 . 7 ∘ . For almost all the NPR for which the nozzle is tested, the boundary layer state is seen to be in the laminar/transitional range and exhibits resonance for nozzle pressure ratios less than 2.3. The resonance tones associated with the laminar interactions result in a considerable increase in the length of the intermittent region. However, as the boundary layer state goes into the transitional regime and the resonance tones disappear, the length of intermittent region is seen to decrease significantly. A decrease in this length also causes the peak standard deviation values to decrease. Further a free interaction region is observed for separated nozzle flows irrespective of the NPR of operation for resonance free interactions.

Shock-Wave / Turbulent Boundary Layer Interactions (SWTBLI) play an important role in optimizations of supersonic and hypersonic inlets. For over 60 years SWTBLIs have been experimentally and numerically studied. Recently a scaling approach for the characteristic interaction length has been described for two dimensional attached and separated flows. It resulted in a best fit law for a variety of flow conditions. In this study this approach is used to scale a data set of several nominal two dimensional SWTBLIs at Mach 3. The data set is the outcome of an intensive study with variable impinging shock strengths. Cylindrical shock generators with a variable slenderness ratio are mounted at two alternative positions over a flat plate. Experiments and computational fluid dynamic (CFD) simulations have been conducted. The experiments were carried out in the Ludwieg Tube Facility at DLR G¨ottingen, employing Pitot probe measurements, mean wall pressure measurements and the high-speed shadowgraph technique. The Reynolds-averaged NavierStokes (RANS) simulations have been validated with literature studies and by comparing them to experimental results. The CFD results are used to characterize the flow topology. A method is proposed to automatically detect the interaction lengths from shadowgrams. They were used as input for the abovementioned scaling approach which produced a best fit correlation for this experimental setup. Differences to the best fit law from the literature occur by decreasing the cylinder slenderness accompanied by three dimensional effects influencing the interaction region.

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.

It is known that the aeroelastic stability of elastic plates significantly decreases in low supersonic flow. A further reduction may arise due to compressive in-plane loads, which can be caused by deformations of the adjacent structure. Thus, this is an important subject in the design of supersonic aircraft or rockets. Encouraged by recent studies that have shown an increase in aerodynamic damping due to a turbulent boundary layer, the present study applies three different aerodynamic models coupled with the von Karman plate equation to show that there is as well a significant enlargement of the static stability regions of a flat plate subjected to quasi-static in-plane loads at Mach 1.2. However, it has also been found that, despite a turbulent boundary layer, the critical dynamic pressure for flutter almost drops to zero if the in-plane load approaches the critical load for Euler buckling. Further analysis on postcritical flutter revealed that a flexible panel subjected to a boundary layer behaves simil...

International audience; Direct numerical simulations (DNS) of Mach = 2.9 supersonic turbulent boundary layerswith spanwise wall oscillation (SWO) are conducted to investigate the turbulent heat transportmechanism and its relation with the turbulent momentum transport. The turbulent coherentstructures are suppressed by SWO and the drag is reduced. Although the velocity and temperaturestatistics are disturbed by SWO differently, the turbulence transports of momentum and heat aresimultaneously suppressed. The Reynolds analogy and the strong Reynolds analogy are alsopreserved in all the controlled flows, proving the consistent mechanisms of momentum transportand heat transport in the turbulent boundary layer with SWO. Despite the extra dissipation andheat induced by SWO, a net wall heat flux reduction can be achieved with the proper selected SWOparameters. The consistent mechanism of momentum and heat transports supports the application ofturbulent drag reduction technologies to wall heat flux controls in high-speed vehicles.

A method for injection of gas into the boundary layer on a slender body in supersonic flow while minimizing perturbation to the mean flow is examined. Injection of gas is equivalent to a sudden increase in the displacement thickness of the boundary layer, which produces an oblique shock that propagates into the inviscid region of the flow. It is found that modification of the geometry of the body can compensate for the increased displacement thickness created by injection and minimize the production of oblique waves. However, the resulting near-wall injection layer is observed to be unstable and a turbulent boundary layer develops downstream of the injection region. The instability of the flow is examined experimentally using high-speed schlieren visualization and numerically using linear stability analysis of velocity profiles from a compressible Navier–Stokes computation. At the present postshock Mach number of about 3.8, both first- and second-mode instabilities are active, though computations predict that the first mode is primarily responsible for transition downstream of the injector.

International audience; A new Dual-particle image velocimetry (Dual-PIV) system for application in supersonic flows was developed. The system was designed for shock wave/turbulent boundary layer interactions with separation. This type of flow places demanding requirements on the system, from the large range of characteristic frequencies O(100 Hz-100 kHz) to spatial and temporal resolutions necessary for the measurement of turbulent quantities (Dolling in AIAA J 39(8): 1517-1531, 2001; Dupont et al. in J Fluid Mech 559: 255-277, 2006; Smits and Dussauge in Turbulent shear layers in supersonic flow, 2nd edn. Springer, New York, 2006). While classic PIV systems using high-resolution CCD sensors allow high spatial resolution, these systems cannot provide the required temporal resolution. Existing high-speed PIV systems provide temporal and CMOS sensor resolutions, and even laser pulse energies, that are not adapted to our needs. The only obvious solution allowing sufficiently high spatial resolution, access to high frequencies, and a high laser pulse energy is a multi-frame system: a Dual-PIV system, consisting of two synchronized PIV systems observing the same field of view, will give access to temporal characteristics of the flow. The key technology of our system is frequency-based image separation: two lasers of different wavelengths illuminate the field of view. The cross-pollution with laser light from the respective other branches was quantified during system validation. The overall system noise was quantified, and the prevailing error of only 2 % reflects the good spatial and temporal alignment. The quality of the measurement system is demonstrated with some results on a subsonic jet flow including the spatio-temporal inter-correlation functions between the systems. First measurements in a turbulent flat-plate boundary layer at Mach 2 show the same satisfactory data quality and are also presented and discussed.

A numerical study, using a mixed finite element/finite volume method on unstructured meshes adapted for compressible flows, is conducted to investigate turbulent boundary-layer separation in overexpanded subscale supersonic nozzles including shock/shock and shock/boundary layer interactions. Two test-cases are investigated, namely a TIC (Truncated Ideal Contour) nozzle and a TOP (Thrust Optimized Parabolic contour) nozzle with a secondary jet injection. Particular attention is paid to the appearance of a recirculation region downstream of the Mach disc as well as in the vicinity of the secondary nozzle exit. The results so obtained are analyzed and compared with the experimental data. The results suggest that very different shock structures and flow separation may appear depending on the nozzle contour as well as on the operating pressure and temperature ratios. In the case of TOP nozzle, the simulations reveal the existence of a small recirculation bubble at the vicinity of the secondary injection, due to a shock/boundary layer interaction. In addition, it has been shown that, at high temperature ratios, compressibility effects on the growth rate of the mixing layer, which develops between the main stream and the secondary jet injection, become significant and cannot be neglected in the computation.

Reducing friction resistance and aerodynamic heating has important engineering significance to improve the performances of super/hypersonic aircraft, so the purpose of transition control and turbulent drag reduction becomes one of the cutting edges in turbulence research. In order to investigate the influences of wall cooling and suction on the transition process and fully developed turbulence, the large eddy simulation of spatially evolving supersonic boundary layer transition over a flat-plate with freestream Mach number 4.5 at different wall temperature and suction intensity is performed in the present work. It is found that the wall cooling and suction are capable of changing the mean velocity profile within the boundary layer and improving the stability of the flow field, thus delaying the onset of the spatial transition process. The transition control will become more effective as the wall temperature decreases, while there is an optimal wall suction intensity under the given conditions. Moreover, the development of large-scale coherent structures can be suppressed effectively via wall cooling, but wall suction has no influence.

The present work is dedicated to the numerical study of statistical characteristics of spatially-evolving supersonic turbulent boundary layers (STBL) with cooled walls. Large-Eddy Simulations (LESs) are performed to gain further insight into the role of wall temperature on the mean and fluctuating-flow properties of STBL. The velocity fluctuations, which are scaled according to the Morkovin’s hypothesis, have shown acceptable agreement with available experimental and DNS results of literature. However, the van Driest transformed skin friction C f inc lies below the incompressible theoretical curves as a function of Re θ inc for cold STBL, whereas compressible skin friction is found to be relatively higher for cold wall boundary layers than adiabatic boundary layers. The variation of total shear stress remains unaffected throughout the boundary layer for 0.5 ⩽ T w / T r ⩽ 1 . The total temperature fluctuations are non-negligible for cold cases and surface cooling changes the near-wall turbulent structures. Additionally, the streamwise velocity and temperature fluctuations for the coldest isothermal STBL case are strongly correlated compared to the anti-correlation behavior of the adiabatic STBL in the near-wall region ( y + ≈ 9 ). Furthermore, the pressure fluctuations are found to be non-negligible for cooled boundary layers and a positive correlation between pressure and density fluctuations are observed in the log-layer. These tendencies have also been verified through a detailed statistical analysis of the unsteady flow-field.

A finite-difference method based on upwind flux difference splitting has been used to predict the performance of supersonic axial compressor cascades and to examine the complex physical flow phenomena. The prediction with Abe, Kondoh, and Nagano’s two-equation turbulence model showed good agreement with the experimental results near the zone of shock-wave/turbulent boundary-layer interaction at different exit-inlet pressure ratios. The increase in exit-inlet pressure ratio caused slight reduction in shock loss but an increase in viscous loss. The passive control of shockwave interaction with turbulent boundary layer by a cavity reduced total pressure losses and strength of shock waves. A cavity under the shock foot point at suction surface of the blades was used as a passive control. The effect of cavity length and depth on total pressure loss was studied. The proper size of cavities could reduce total pressure losses even though it increased viscous losses through the formation of shock-wave/turbulent boundary-layer zone.

Selection and sizing of the thermal protection material (TPM) are the two key performance parameters in TPS design, and prediction inaccuracies can be fatal for the crew or the success of robotic missions. Ground testing in plasma wind-tunnels becomes a fundamental requirement for validation of material response codes, TPM qualification, and TPS design. Although present wind-tunnel technology does not allow a complete simulation of the hypersonic flight, specific phenomena can be studied separately in a narrowed environment, where dominant parameters, such as heat flux, pressure and the resulting boundary layer chemistry are close to the reentry environment. Ablation experiments have been carried out with the carbon-phenolic material AQ61 and a non-pyrolyzing carbon fiber preform in sub- and supersonic air and nitrogen plasmas in the VKI Plasmatron facility. We performed an in-situ recession analysis, including volumetric ablation, and observed the temperature and radiance of the surface for emissivity estimations, as well as spatial molecular radiation profiles in the boundary layer, which were compared to a numerical approach. Surface temperatures as low as 1600K allowed observation of the temperature distribution over the surface using an infrared camera. The carbon preform test samples were varied in shape as well as regarding their fiber direction. Sublimation of the surface was reached in supersonic plasma flow at a cold wall heat flux of 9.5MW/m2. Carbon preform emissivities were found to be in the order of 0.86 - 0.97. A stagnation line description with an ablation boundary condition helped to reproduce experimentally measured boundary layer emission profiles of the CN violet molecule. In spite of the very simplified model, emission intensities were in the same order of magnitude compared to the experimental data. The model was able to reconstruct the location of the maximum emission for two cases.

We report on laboratory experiments examining the interaction of a jet from an overpressurized reservoir with a canonical ground surface to simulate lateral blasts at volcanoes such as the 1980 blast at Mount St. Helens. These benchmark experiments test the application of supersonic jet models to simulate the flow of volcanic jets over a lateral topography. The internal shock structure of the free jet is modified such that the Mach disk shock is elevated above the surface. In elevation view, the width of the shock is reduced in comparison with a free jet, while in map view the dimensions are comparable. The distance of the Mach disk shock from the vent is in good agreement with free jet data and can be predicted with existing theory. The internal shock structures can interact with and penetrate the boundary layer. In the shock-boundary layer interaction, an oblique shock foot is present in the schlieren images and a distinctive ground signature is evident in surface measurements. The location of the oblique shock foot and the surface demarcation are closely correlated with the Mach disk shock location during reservoir depletion, and therefore, estimates of a ground signature in a zone devastated by a blast can be based on the calculated shock location from free jet theory. These experiments, combined with scaling arguments, suggest that the imprint of the Mach disk shock on the ground should be within the range of 4–9 km at Mount St. Helens depending on assumed reservoir pressure and vent dimensions.

Implicitly implemented large eddy simulation (LES) with a fifth-order WENO scheme was conducted in this study. Based on Navier-Stokes equations, this LES was carried out to explore the origin of momentum deficit caused by a supersonic micro-ramp at flow conditions of M a = 2.5 and Re θ = 5760. The numerical results were validated through qualitative and quantitative comparisons with existing experimental data. After describing the aerodynamic properties of the supersonic wake, such as the deficit and the streamwise vortices, the momentum deficit was later detected to originate from the lower portion of the upstream boundary layer, while the high momentum fluid originated from close to the wall at the upper portion. Position alternation trigged by the micro-ramp was finally proposed as a revised mechanism.

The large eddy simulation (LES) of spatially evolving supersonic boundary layer transition over a flat-plate with freestream Mach number 4.5 is performed in the present work. The Favre-filtered Navier-Stokes equations are used to simulate large scales, while a dynamic mixed subgrid-scale (SGS) model is used to simulate subgrid stress. The convective terms are discretized with a fifth-order upwind compact difference scheme, while a sixth-order symmetric compact difference scheme is employed for the diffusive terms. The basic mean flow is obtained from the similarity solution of the compressible laminar boundary layer. In order to ensure the transition from the initial laminar flow to fully developed turbulence, a pair of oblique first-mode perturbation is imposed on the inflow boundary. The whole process of the spatial transition is obtained from the simulation. Through the space-time average, the variations of typical statistical quantities are analyzed. It is found that the distributions of turbulent Mach number, root-mean-square (rms) fluctuation quantities, and Reynolds stresses along the wall-normal direction at different streamwise locations exhibit self-similarity in fully developed turbulent region. Finally, the onset and development of large-scale coherent structures through the transition process are depicted.

The flow past a microramp immersed in a supersonic turbulent boundary layer is studied by means of numerical simulations with the implicit large-eddy simulation technique and experiments conducted with tomographic particle image velocimetry. The experimental data are mostly used to verify the validity of the numerical results by ample comparisons on the time-averaged velocity, turbulent statistics, and vortex intensity. Although some discrepancies are observed on the intensity of the upwash motion generated by the streamwise vortex pair, the rates of the recovery of momentum deficit and the decay of streamwise vortex pair intensity are found in good agreement. The instantaneous flow organization is inspected, making use of the flow realizations available from implicit large-eddy simulation. The flow behind the microramp exhibits significant large-scale unsteady fluctuations. Notably, the quasi-conical shear layer enclosing the wake is strongly undulated under the action of Kelvin–Helmholtz (K–H) vortices. The resulted vortices induce localized high-speed arches in the outer region and a deceleration within the wake associated with ejection of low-momentum fluid. Because of the presence of the K–H vortex, the streamwise vortex filaments exhibit downward and outward motions. The further evolution of vortical structures within the wake features the development of K–H vortices from arch shape to full ring in the far wake, under the effects of the streamwise vortices, which induce an inward motion of the vortex legs and eventually connect the vortex at the bottom.

High pressure turbine stages work in transonic regimes and then shock waves, shed by the trailing edge, impinge on the suction side modifying the flow structures. Gas turbine entry temperature is much higher than the allowable material limit and the hot components can survive only using advanced film-cooling systems. Unfortunately these systems are designed without taking into account the interaction with the shock waves and this article would like to address this problem and to evaluate if this assumption is correct or not. A correct prediction and understanding of the interaction between the ejected coolant and the shock waves is crucial in order to achieve an optimal distribution of the coolant and to increase the components’ life. In this work, the numerical investigation of a film-cooling test case, investigated experimentally by the University of Karlsruhe, is shown. An in-house computational fluid dynamics solver is used for the numerical analysis. The test rig consists of a converging–diverging nozzle that accelerates the incoming flow up to supersonic conditions and an oblique shock is generated at the nozzle exit section. Three cases have been studied, where the cooling holes have been positioned before, near and after the shock impingement. The results obtained considering four blowing ratios are presented and compared with the available experimental data. The local adiabatic effectiveness is affected by the shock–coolant interaction and this effect has been observed for all the blowing ratios investigated. A similar trend is observed in the experimental data even if the numerical simulations over-predict the impact of the interaction.

Boundary layers constitute a fundamental source of aerodynamic noise. A turbulent boundary layer over a plane wall can provide an indirect contribution to the noise by exciting the structure and a direct noise contribution. The latter part can play a significant role even if its intensity is very low, explaining why it is difficult to measure. In the present study, the aerodynamic noise generated by a spatially developing turbulent boundary layer is computed directly by solving the compressible Navier–Stokes equations. This numerical experiment aims at giving some insight into the noise radiation characteristics. The acoustic wavefronts have a large wavelength and are oriented in the direction opposite to the flow. Their amplitude is only 0.7 % of the aerodynamic pressure for a flat-plate flow at Mach 0.5. The particular directivity is mainly explained by convection effects by the mean flow, giving an indication about the compactness of the sources. These vortical events correspond to low frequencies and thus have a large lifetime. They cannot be directly associated with the main structures populating the boundary layer such as hairpin or horseshoe vortices. The analysis of the wall pressure can provide a picture of the flow in the wavenumber–frequency space. The main features of wall pressure beneath a turbulent boundary layer as described in the literature are well reproduced. The acoustic domain, corresponding to supersonic wavenumbers, is detectable but can hardly be separated from the convective ridge at this relatively high speed. This is also due to the low frequencies of sound emission as noted previously.

Flows of relevance to new generation aerospace vehicles exist, which are weakly dependent on the streamwise direction and strongly dependent on the other two spatial directions, such as the flow around the (flattened) nose of the vehicle and the associated elliptic cone model. Exploiting these characteristics, a parabolic integration of the Navier-Stokes equations is more appropriate than solution of the full equations, resulting in the so-called Parabolic Navier-Stokes (PNS). This approach not only is the best candidate, in terms of computational efficiency and accuracy, for the computation of steady base flows with the appointed properties, but also permits performing instability analysis and laminar-turbulent transition studies a-posteriori to the base flow computation. This is to be contrasted with the alternative approach of using order-of-magnitude more expensive spatial Direct Numerical Simulations (DNS) for the description of the transition process. The PNS equations used here have been formulated for an arbitrary coordinate transformation and the spatial discretization is performed using a novel stable high-order finite-difference-based numerical scheme, ensuring the recovery of highly accurate solutions using modest computing resources. For verification purposes, the boundary layer solution around a circular cone at zero angle of attack is compared in the incompressible limit with theoretical profiles. Also, the recovered shock wave angle at supersonic conditions is compared with theoretical predictions in the same circular-base cone geometry. Finally, the entire flow field, including shock position and compressible boundary layer around a 2:1 elliptic cone is recovered at Mach numbers 3 and 4

This article presents the computation of a set of control parameters for the deterministic prediction of laminar boundary-layer instabilities induced by an imposed unsteady shock interaction. The objective of the study is exploratory in nature by computing a supersonic flight environment for flow over a blunt body and the deterministic prediction of the spectral entropy rates for the boundary layer subjected to an unsteady pressure disturbance. The deterministic values for the spectral entropy rate within the instabilities are determined for each control parameter. Computational results imply that the instabilities are of a span-wise vortex form, that the maximum vertical velocity wave vector components are produced in the region nearest the wall and that extended transient coherent structures are produced in the boundary layer at a vertical location slightly below the mid-point of the boundary layer.

The aim of this work is to analyze a viscous flow in the axisymmetric nozzle taken into account the mesh size both in the free stream and into the boundary layer. The resolution of the Navier- Stokes equations is realized by using the finite volume method to determine the supersonic flow parameters at the exit of convergingdiverging nozzle. The numerical technique uses the Flux Vector Splitting method of Van Leer. Here, adequate time stepping parameter, along with CFL coefficient and mesh size level is selected to ensure numerical convergence. The effect of the boundary layer thickness is significant at the exit of the nozzle. The best solution is obtained with using a very fine grid, especially near the wall, where we have a strong variation of velocity, temperature and shear stress. This study enabled us to confirm that the determination of boundary layer thickness can be obtained only if the size of the mesh is lower than a certain value limits given by our calculations., {"references":["Goudjo, J.A. Désidéri, \" a finite volume scheme to resolution an\naxisymmetric Euler equations (Un schéma de volumes finis décentré\npour la résolution des équations d-Euler en axisymétrique), \" Research\nreport INRIA 1005, 1989.","R. Haoui, A. Gahmousse, D. Zeitoun, \" Chemical and vibrational\nnonequilibrium flow in a hypersonic axisymmetric nozzle, \"\nInternational Journal of Thermal Sciences, article n┬░ 8 , volume 40,\n(2001), pp787-795.","R. Haoui, \"Finite volumes analysis of a supersonic non-equilibrium\nflow around the axisymmetric blunt body,\" International Journal of\nAeronautical and space Sciences, 11(2), (2010), pp59-68.","R. Haoui,\" Application of the finite volume method for the supersonic\nflow around the axisymmetric cone body placed in free stream,\" WIT\npress, Fourteenth international conference on computational methods\nand experimental measurements, pp379-388, Southampton, 2009.","B. Van Leer, \"Flux Vector Splitting for the Euler Equations,\" Lecture\nNotes in Physics. 170, (1982), 507-512.","R. Haoui, A. Gahmousse, D. Zeitoun, \"Condition of convergence\napplied to an axisymmetric reactive flow,\" 16th CFM, n┬░738, Nice,\nFrance, 2003.","H. Schlichting, \"Boundary-layer theory,\" 7th edition, McGraw-Hill,\nNew York, 1979.","K. A. Hoffmann, \"Computational fluid dynamics for engineers,\"\nVolume II. Chapter 14, Engineering Education system, Wichita, USA,\npp.202-235, 1995.","J.H. Ferziger & all, \"Computational Methods for Fluid Dynamics,\"\nChapter 8, Springer-Verlag, Berlin Heidelberg, New York, pp.217-259,\n2002."]}