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A. Daliri Rezagholigheshlaghi
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1
A technique that generalizes pressure reconstruction from planar and volumetric particle image velocimetry (PIV) velocity data is proposed to eliminate the need for a user-specified choice of boundary conditions prior to pressure gradient integration. The method extends the dual model concept proposed by Jux et al (2020 Meas. Sci. Technol. 31 104001), which partitions the measurement domain into irrotational (IRR) and rotational (ROT) regions using the total pressure coefficient as a discriminant. The latter approach requires an educated choice of the IRR boundary condition location as well as a threshold (minimum) value set for the total pressure coefficient in the IRR region. Iterative domain partitioning (IDP) approaches the problem by evaluating IRR and ROT iteratively and thereby significantly reducing the result sensitivity to the user’s initial choice of IRR boundary and threshold value. Furthermore, a near-wall adaptation of the integration, based on iso-potential lines, is introduced to avoid excessive error propagation when integrating PIV data along the object boundaries. The method is demonstrated to reliably converge towards a unique IRR-ROT boundary and minimize error propagation. The verification is carried out by examining three-dimensional (3D) particle tracking velocimetry data, reduced to planar, from the steady flow over a NACA0015 airfoil at α = [0°, 10°, 20°]. Monte Carlo simulation with randomly varying boundary conditions yields the lowest dispersion of pressure distribution, as a metric to represent the method’s accuracy. The approach is generalized to data from 3D PIV around a wall-mounted cube where the surface pressure reconstruction reveals a distribution consistent with the flow field topology. Also, in 3D, variation of the boundary conditions yields minimal fluctuations of the reconstructed pressure field when using IDP.
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A technique that generalizes pressure reconstruction from planar and volumetric particle image velocimetry (PIV) velocity data is proposed to eliminate the need for a user-specified choice of boundary conditions prior to pressure gradient integration. The method extends the dual model concept proposed by Jux et al (2020 Meas. Sci. Technol. 31 104001), which partitions the measurement domain into irrotational (IRR) and rotational (ROT) regions using the total pressure coefficient as a discriminant. The latter approach requires an educated choice of the IRR boundary condition location as well as a threshold (minimum) value set for the total pressure coefficient in the IRR region. Iterative domain partitioning (IDP) approaches the problem by evaluating IRR and ROT iteratively and thereby significantly reducing the result sensitivity to the user’s initial choice of IRR boundary and threshold value. Furthermore, a near-wall adaptation of the integration, based on iso-potential lines, is introduced to avoid excessive error propagation when integrating PIV data along the object boundaries. The method is demonstrated to reliably converge towards a unique IRR-ROT boundary and minimize error propagation. The verification is carried out by examining three-dimensional (3D) particle tracking velocimetry data, reduced to planar, from the steady flow over a NACA0015 airfoil at α = [0°, 10°, 20°]. Monte Carlo simulation with randomly varying boundary conditions yields the lowest dispersion of pressure distribution, as a metric to represent the method’s accuracy. The approach is generalized to data from 3D PIV around a wall-mounted cube where the surface pressure reconstruction reveals a distribution consistent with the flow field topology. Also, in 3D, variation of the boundary conditions yields minimal fluctuations of the reconstructed pressure field when using IDP.
Journal article
(2023)
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Abbas Daliri , Mohammad Javad Maghrebi, Mohammad Reza Soltani
Hot-film sensors have been used to measure the boundary-layer fluctuations and subsequently to determine the boundary-layer states. In this paper, it is proposed to use hot-film sensors as a tool to measure the effects of plasma-induced ionic wind on the boundary-layer fluctuations which can be used to find the influence of the plasma actuator far downstream of the actuator. Time history and frequency response of the hot-film output voltage are investigated to explore the effects of plasma-induced wind downstream of the actuator. Also, the effects of the peak-to-peak voltage of a DBD plasma actuator on the fluctuations of the hot-film output voltages are discussed. It is showed that as far as the effects of ionic wind can be sensed by the hot-films, the control authority of the actuator is higher. It is concluded that induced velocity in the vicinity of the plasma actuator alone does not indicate the control authority of the plasma actuator.
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Hot-film sensors have been used to measure the boundary-layer fluctuations and subsequently to determine the boundary-layer states. In this paper, it is proposed to use hot-film sensors as a tool to measure the effects of plasma-induced ionic wind on the boundary-layer fluctuations which can be used to find the influence of the plasma actuator far downstream of the actuator. Time history and frequency response of the hot-film output voltage are investigated to explore the effects of plasma-induced wind downstream of the actuator. Also, the effects of the peak-to-peak voltage of a DBD plasma actuator on the fluctuations of the hot-film output voltages are discussed. It is showed that as far as the effects of ionic wind can be sensed by the hot-films, the control authority of the actuator is higher. It is concluded that induced velocity in the vicinity of the plasma actuator alone does not indicate the control authority of the plasma actuator.
Journal article
(2023)
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Abbas Daliri
The accuracy of Theodorsen's lift model for pure-pitch, pure-plunge and combined pitch-plunge oscillations of a two-dimensional model is compared with wind tunnel results. The reduced frequency of the oscillation was in the range of 0.01 < k < 0.15, and the freestream Reynolds number was in the range of 130 000 < R e < 400 000. The lift response to an uncoupled combined pitch-plunge motion (for which the frequency of pitch and plunge motions were not the same) is discussed using experimental results. The Theodorsen's lift model is rewritten for the general uncoupled pitch-plunge motions by a linear superposition of all components of the airfoil bound circulation. Both amplitude and phase from the Theodorsen's function are compared with those of the wind tunnel data, and the results are discussed. The Theodorsen's function is found to be a good estimator for both pure-pitch and pure-plunge motions. It further appropriately estimates the lift amplitude for the case of coupled pitch-plunge motion; however, the prediction is not accurate for the uncoupled pitch-plunge motion. A motion amplitude ratio is defined, which shows the level of aperiodicity of the motion. Discrepancy between experimental and analytical results increase with the reduction of the lift amplitude ratio and with the deviation of frequency ratio from unity.
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The accuracy of Theodorsen's lift model for pure-pitch, pure-plunge and combined pitch-plunge oscillations of a two-dimensional model is compared with wind tunnel results. The reduced frequency of the oscillation was in the range of 0.01 < k < 0.15, and the freestream Reynolds number was in the range of 130 000 < R e < 400 000. The lift response to an uncoupled combined pitch-plunge motion (for which the frequency of pitch and plunge motions were not the same) is discussed using experimental results. The Theodorsen's lift model is rewritten for the general uncoupled pitch-plunge motions by a linear superposition of all components of the airfoil bound circulation. Both amplitude and phase from the Theodorsen's function are compared with those of the wind tunnel data, and the results are discussed. The Theodorsen's function is found to be a good estimator for both pure-pitch and pure-plunge motions. It further appropriately estimates the lift amplitude for the case of coupled pitch-plunge motion; however, the prediction is not accurate for the uncoupled pitch-plunge motion. A motion amplitude ratio is defined, which shows the level of aperiodicity of the motion. Discrepancy between experimental and analytical results increase with the reduction of the lift amplitude ratio and with the deviation of frequency ratio from unity.
Doctoral thesis
(2022)
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Abbas Daliri
Journal article
(2021)
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Abbas Daliri , Mohmmad Javad Maghrebi, Mohammad Reza Soltani
The boundary-layer control authority of a DBD plasma actuator using surface mounted hot-film sensors is evaluated. Wind tunnel experiments on a wind-turbine blade section were established at a Reynolds number of 0.27× 106. Aerodynamic performance of the wind-turbine blade section for both plasma-ON and plasma-OFF modes are evaluated using measurements made by both surface pressure and wake survey behind the model. Two distinct boundary-layer states are recognized. A state which occurs at the onset and in proximity of the deep stall, which is affected by the low-frequency instabilities of the separated flow. In this case, the steady actuation of plasma imparts local momentum on the nearby flow, eliminating the instabilities, hence, reattaching the detached flow. The other state happens beyond the static stall angle of attack of the airfoil where the flow over the suction side of the airfoil is fully separated and coexistence of both the leading edge and the trailing edge shear-layer instabilities and natural trailing edge vortex shedding is the underlying mechanism. In this case, although the plasma actuator eliminates the instabilities, to some extent, but the corresponding momentum injection is not efficient to stabilize and reattach the flow.
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The boundary-layer control authority of a DBD plasma actuator using surface mounted hot-film sensors is evaluated. Wind tunnel experiments on a wind-turbine blade section were established at a Reynolds number of 0.27× 106. Aerodynamic performance of the wind-turbine blade section for both plasma-ON and plasma-OFF modes are evaluated using measurements made by both surface pressure and wake survey behind the model. Two distinct boundary-layer states are recognized. A state which occurs at the onset and in proximity of the deep stall, which is affected by the low-frequency instabilities of the separated flow. In this case, the steady actuation of plasma imparts local momentum on the nearby flow, eliminating the instabilities, hence, reattaching the detached flow. The other state happens beyond the static stall angle of attack of the airfoil where the flow over the suction side of the airfoil is fully separated and coexistence of both the leading edge and the trailing edge shear-layer instabilities and natural trailing edge vortex shedding is the underlying mechanism. In this case, although the plasma actuator eliminates the instabilities, to some extent, but the corresponding momentum injection is not efficient to stabilize and reattach the flow.
Journal article
(2021)
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Hassan Akhlaghi, Ehsan Roohi, Abbas Daliri , Mohammad Reza Soltani
Well-known polars in classical shock wave theory, that is, flow deflection angle-shock angle (θ-β), hodograph (u*,v*), and pressure deflection (θ-P*) diagrams, are investigated for the rarefied gas flows using a recently proposed shock wave detection technique by Akhlaghi and coworkers. The agreement between the obtained polars with the analytical relations in classical shock wave theory has been shown in the continuum limit for the cases of supersonic flow over the wedge and cylinder geometries. Investigations are performed using the RGS2D direct simulation Monte Carlo solver for supersonic gas flows over a circular cylinder at continuum limit and Kn = 10−4, 10−3, 0.01, 0.03, 0.07, and 0.10. Two species of nitrogen and argon at various Mach numbers of 1.5, 3.0, and 10.0 are considered. The shock polars are investigated along bow shock waves in front of the cylinder. The results indicate that rarefaction significantly affects the shock polars. As Knudsen number increases, shock angle, maximum flow deflection angle, and aft shock pressure increase. However, velocity components after the shock wave decrease as the flow becomes more rarefied. These effects are stronger for θ-β polar under the weak shock condition. Meanwhile, they are stronger for θ-P* and hodograph polars in strong shock situations.
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Well-known polars in classical shock wave theory, that is, flow deflection angle-shock angle (θ-β), hodograph (u*,v*), and pressure deflection (θ-P*) diagrams, are investigated for the rarefied gas flows using a recently proposed shock wave detection technique by Akhlaghi and coworkers. The agreement between the obtained polars with the analytical relations in classical shock wave theory has been shown in the continuum limit for the cases of supersonic flow over the wedge and cylinder geometries. Investigations are performed using the RGS2D direct simulation Monte Carlo solver for supersonic gas flows over a circular cylinder at continuum limit and Kn = 10−4, 10−3, 0.01, 0.03, 0.07, and 0.10. Two species of nitrogen and argon at various Mach numbers of 1.5, 3.0, and 10.0 are considered. The shock polars are investigated along bow shock waves in front of the cylinder. The results indicate that rarefaction significantly affects the shock polars. As Knudsen number increases, shock angle, maximum flow deflection angle, and aft shock pressure increase. However, velocity components after the shock wave decrease as the flow becomes more rarefied. These effects are stronger for θ-β polar under the weak shock condition. Meanwhile, they are stronger for θ-P* and hodograph polars in strong shock situations.
Journal article
(2020)
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Hoda Mahdavi, Abbas Daliri , Farshad Sohbatzadeh, Meisam Shirzadi, Mohammad Rezanejad
Dielectric barrier discharge plasma actuators have attracted a lot of attention to use as new technologies for active flow control. In this paper, an experimental investigation of a single unsteady plasma actuator driven by two simultaneous sinusoidal high voltages is carried out. The aerodynamic performance of this plasma actuator is investigated for flow control of a wind turbine blade. Leading edge separation control at Reynolds number of 0.26 × 10 6 and in a wide range of angles of attack including linear, stall, and post-stall regions is considered as the test case. The momentum imparted by the plasma actuator to flow is investigated via measuring induced electric wind velocity, which represents that the induced velocities in the steady mode, with v max ≅ 5 m / s, are higher than that of the unsteady mode. The important aspect of exciting the unsteady dielectric barrier discharge plasma actuator in this new approach is improving its efficiency through increasing the authority of momentum addition to flow and reducing the minimum input power for discharge ignition relative to its typical grounded structure. This can be very important practically in flow control applications where the amount of consumed energy is a substantial factor in determining the actuator's efficiency. The obtained aerodynamic results reveal that the unsteady plasma actuator has the best operation in post-stall angles of attack, which is of great importance for operation characteristics of wind turbine blades. The power spectral density of pressure time series illustrates that the unsteady plasma actuator affects the flow through instabilities of the separation layer and vortex shedding structure.
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Dielectric barrier discharge plasma actuators have attracted a lot of attention to use as new technologies for active flow control. In this paper, an experimental investigation of a single unsteady plasma actuator driven by two simultaneous sinusoidal high voltages is carried out. The aerodynamic performance of this plasma actuator is investigated for flow control of a wind turbine blade. Leading edge separation control at Reynolds number of 0.26 × 10 6 and in a wide range of angles of attack including linear, stall, and post-stall regions is considered as the test case. The momentum imparted by the plasma actuator to flow is investigated via measuring induced electric wind velocity, which represents that the induced velocities in the steady mode, with v max ≅ 5 m / s, are higher than that of the unsteady mode. The important aspect of exciting the unsteady dielectric barrier discharge plasma actuator in this new approach is improving its efficiency through increasing the authority of momentum addition to flow and reducing the minimum input power for discharge ignition relative to its typical grounded structure. This can be very important practically in flow control applications where the amount of consumed energy is a substantial factor in determining the actuator's efficiency. The obtained aerodynamic results reveal that the unsteady plasma actuator has the best operation in post-stall angles of attack, which is of great importance for operation characteristics of wind turbine blades. The power spectral density of pressure time series illustrates that the unsteady plasma actuator affects the flow through instabilities of the separation layer and vortex shedding structure.
Journal article
(2019)
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Mohammad Farahani, Abbas Daliri , Javad Sepahi Younsi
Feasibility of an innovative buzz detection technique through measuring the static pressure outside a mixed-compression supersonic inlet is studied. The buzz is an instability phenomenon that occurs almost in all supersonic inlets. During the buzz, shock oscillation along with pressure and mass flow fluctuations affects the performance characteristics of the inlet. The main objective of this paper is to introduce a simple and easy-to-implement method for investigation of the buzz phenomenon in a supersonic inlet. The experimental data for far field-based are compared with those of the model-based one at free stream Mach numbers of 1.8, 2.0, and 2.2 and at zero degrees angle of attack for a mixed-compression inlet. The results show that this technique can measure exact value of buzz frequency as well as its onset. The present method uses pressure data obtained from the wind-tunnel wall instead of measuring the pressure inside or on the model surfaces which in most cases is very hard. However, to sense flow oscillations, caused by the buzz onset the sensors on the wind-tunnel wall must be located downstream of the point where the oblique shock from the spike impinges on the tunnel wall. The location of the measurement point as well as the distance between the sensor and the origin of the shock wave system are very important.
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Feasibility of an innovative buzz detection technique through measuring the static pressure outside a mixed-compression supersonic inlet is studied. The buzz is an instability phenomenon that occurs almost in all supersonic inlets. During the buzz, shock oscillation along with pressure and mass flow fluctuations affects the performance characteristics of the inlet. The main objective of this paper is to introduce a simple and easy-to-implement method for investigation of the buzz phenomenon in a supersonic inlet. The experimental data for far field-based are compared with those of the model-based one at free stream Mach numbers of 1.8, 2.0, and 2.2 and at zero degrees angle of attack for a mixed-compression inlet. The results show that this technique can measure exact value of buzz frequency as well as its onset. The present method uses pressure data obtained from the wind-tunnel wall instead of measuring the pressure inside or on the model surfaces which in most cases is very hard. However, to sense flow oscillations, caused by the buzz onset the sensors on the wind-tunnel wall must be located downstream of the point where the oblique shock from the spike impinges on the tunnel wall. The location of the measurement point as well as the distance between the sensor and the origin of the shock wave system are very important.
Journal article
(2018)
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Abbas Daliri , Mohammad Farahani, Javad Sepahi Younsi
Journal article
(2017)
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Hassan Skhlaghi, Abbas Daliri , Mohammad Reza Soltani
This paper introduces a shock-wave-detection technique based on the schlieren imaging for continuum and rarefied-gas flows. The scheme is applicable for any existing two-dimensional flowfields obtained by experimental or numerical approaches. A Gaussian distribution for a schlieren function within the shock-wave region is considered. This enables the authors to access any desired locations through the shock (e.g., shock center, or leading- and trailing-edge locations). The bow shock-wave profile is described via a rational function, which could be employed for the estimation of shock angle. The relation between pre- and postshock flow properties along the shock wave with a high resolution can be investigated by using this technique. The technique is verified in a novel way based on the well-known gas dynamics curves (i.e., flow-deflection angle vs shock angle and shock polar diagrams). High-speed continuum, near continuum, and rarefied-gas flows over a wedge and cylinder are considered for evaluation of the proposed technique. The results show a very good agreement with existing analytical relations for continuum flow.
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This paper introduces a shock-wave-detection technique based on the schlieren imaging for continuum and rarefied-gas flows. The scheme is applicable for any existing two-dimensional flowfields obtained by experimental or numerical approaches. A Gaussian distribution for a schlieren function within the shock-wave region is considered. This enables the authors to access any desired locations through the shock (e.g., shock center, or leading- and trailing-edge locations). The bow shock-wave profile is described via a rational function, which could be employed for the estimation of shock angle. The relation between pre- and postshock flow properties along the shock wave with a high resolution can be investigated by using this technique. The technique is verified in a novel way based on the well-known gas dynamics curves (i.e., flow-deflection angle vs shock angle and shock polar diagrams). High-speed continuum, near continuum, and rarefied-gas flows over a wedge and cylinder are considered for evaluation of the proposed technique. The results show a very good agreement with existing analytical relations for continuum flow.
Journal article
(2016)
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Mohammad Reza Soltani, Abbas Daliri , Javad Sepahi Younsi, Mohammad Farahani
Effects of various boundary-layer bleed locations on the stability of an axisymmetric supersonic inlet has been investigated experimentally. The bleeds were located on the inlet compression cone and were designed to improve the stability margin of the inlet. The main objective of the study was to identify the boundary-layer bleed location and its effects on the improvement of the stability margin of the inlet. In addition, a buzz precursor detection methodology based on the rms level has been introduced. Experiments have been carried out on a mixed-compression inlet with a design Mach number of 2.0 and three different bleed locations at three freestream Mach numbers of 1.8, 2.0, and 2.2. All tests were conducted at an angle of attack of 0 deg. The results show that, based on the individual design of an inlet, a location can be found that the spilled normal shock stands at this location and consequently separation starts at the buzz onset. If the bleed slot is located in this place, the subcritical performance and stability margin of the inlet can be enhanced.
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Effects of various boundary-layer bleed locations on the stability of an axisymmetric supersonic inlet has been investigated experimentally. The bleeds were located on the inlet compression cone and were designed to improve the stability margin of the inlet. The main objective of the study was to identify the boundary-layer bleed location and its effects on the improvement of the stability margin of the inlet. In addition, a buzz precursor detection methodology based on the rms level has been introduced. Experiments have been carried out on a mixed-compression inlet with a design Mach number of 2.0 and three different bleed locations at three freestream Mach numbers of 1.8, 2.0, and 2.2. All tests were conducted at an angle of attack of 0 deg. The results show that, based on the individual design of an inlet, a location can be found that the spilled normal shock stands at this location and consequently separation starts at the buzz onset. If the bleed slot is located in this place, the subcritical performance and stability margin of the inlet can be enhanced.
Journal article
(2016)
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Mohammad Reza Soltani, Abbas Daliri , Javad Sepahi Younsi
Experiments were conducted to study various kinds of shock wave/boundary layer interaction (SBLI) in an axisymmetric mixed-compression inlet. Further, some of the experimental findings were compared and verified by numerical solutions where possible. A classification of different types of SBLI relevant to the mixed-compression inlets is performed. Interactions of expelled normal shock wave/boundary-layer at subcritical and at buzz condition is investigated using Schlieren and shadowgraph flow visualization as well as unsteady pressure recordings. The data is further compared with the CFD prediction. Interactions of cowl lip reflected oblique shock and the terminal normal shock with the spike boundary-layer at both critical and supercritical operations that leads to shock trains and pseudo-shock phenomena are also studied. In this case numerical simulation results were used to illustrate the flow field. Experimental pressure recordings are used for validation and further discussion. The structure of SBLI flow in an inlet depends highly on the throttling value. For near critical throttling values, interaction of internal compression oblique shocks with boundary-layer and pseudo-shock phenomenon is dominant. Increasing the inlet back pressure pushes normal shock wave out of the inlet duct. Formation of lambda shock due to interaction of separated boundary-layer with normal shock wave is also investigated. The numerical and experimental results show that there exist different kinds of shock wave boundary-layer interactions relevant to the supersonic inlets. Each of these flow interaction phenomena has different effects on the stability and on the performance of the inlet. Interaction of terminal normal shock with internal duct boundary-layer causes pseudo-shock phenomenon that leads to increase of flow distortion and reduction of total pressure recovery. In addition interaction of normal shock wave with external cone boundary-layer causes buzz instability and degrades inlet performance.
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Experiments were conducted to study various kinds of shock wave/boundary layer interaction (SBLI) in an axisymmetric mixed-compression inlet. Further, some of the experimental findings were compared and verified by numerical solutions where possible. A classification of different types of SBLI relevant to the mixed-compression inlets is performed. Interactions of expelled normal shock wave/boundary-layer at subcritical and at buzz condition is investigated using Schlieren and shadowgraph flow visualization as well as unsteady pressure recordings. The data is further compared with the CFD prediction. Interactions of cowl lip reflected oblique shock and the terminal normal shock with the spike boundary-layer at both critical and supercritical operations that leads to shock trains and pseudo-shock phenomena are also studied. In this case numerical simulation results were used to illustrate the flow field. Experimental pressure recordings are used for validation and further discussion. The structure of SBLI flow in an inlet depends highly on the throttling value. For near critical throttling values, interaction of internal compression oblique shocks with boundary-layer and pseudo-shock phenomenon is dominant. Increasing the inlet back pressure pushes normal shock wave out of the inlet duct. Formation of lambda shock due to interaction of separated boundary-layer with normal shock wave is also investigated. The numerical and experimental results show that there exist different kinds of shock wave boundary-layer interactions relevant to the supersonic inlets. Each of these flow interaction phenomena has different effects on the stability and on the performance of the inlet. Interaction of terminal normal shock with internal duct boundary-layer causes pseudo-shock phenomenon that leads to increase of flow distortion and reduction of total pressure recovery. In addition interaction of normal shock wave with external cone boundary-layer causes buzz instability and degrades inlet performance.
Journal article
(2015)
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Mohammad Reza Soltani, Abbas Daliri , Javad Sepahi Younsi
The performance of a supersonic mixed compression air intake has been investigated experimentally. The intake is of axisymmetric one and has been designed for a free stream Mach number of 2.0. The present work has two main goals, first to investigate the performance of the intake without boundary layer bleed at design and at off-design conditions and second to study the effects of a bleed slot on the intake performance. The intake has been tested at free stream Mach numbers of 1.8, 2.0, and 2.2 and at zero degrees angle of attack. Total pressure recovery, mass flow ratio, and flow distortion have been selected to assess the intake performance. The bleed slot is located upstream of the intake throat perpendicular to the compression ramp surface. The suction is applied by the natural pressure difference between the entrance and the exit faces of the bleed duct. Results show that applying the boundary layer suction upstream of the intake throat can considerably improve the intake performance at its design and off-design conditions while it does not affect the intake mass flow rate.
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The performance of a supersonic mixed compression air intake has been investigated experimentally. The intake is of axisymmetric one and has been designed for a free stream Mach number of 2.0. The present work has two main goals, first to investigate the performance of the intake without boundary layer bleed at design and at off-design conditions and second to study the effects of a bleed slot on the intake performance. The intake has been tested at free stream Mach numbers of 1.8, 2.0, and 2.2 and at zero degrees angle of attack. Total pressure recovery, mass flow ratio, and flow distortion have been selected to assess the intake performance. The bleed slot is located upstream of the intake throat perpendicular to the compression ramp surface. The suction is applied by the natural pressure difference between the entrance and the exit faces of the bleed duct. Results show that applying the boundary layer suction upstream of the intake throat can considerably improve the intake performance at its design and off-design conditions while it does not affect the intake mass flow rate.