Events in turbulent flows computed by direct numerical simulation (DNS) are often calibrated with... more Events in turbulent flows computed by direct numerical simulation (DNS) are often calibrated with properties based on homogeneous isotropic turbulence, advanced by Kolmogorov, and given in Turbulence, U. Frisch, Cambridge Univ. Press, UK (1995). However, these computational procedures are not calibrated using numerical analyses in order to assess their strengths and weaknesses for DNS. This is with the exception in "A critical assessment of simulations for transitional and turbulence flows- Sengupta, T.K., In Proc. of IUTAM Symp. on Advances in Computation, Modeling and Control of Transitional and Turbulent Flows, 491-532, (eds. Sengupta, Lele, Sreenivasan, Davidson), WSPC, Singapore (2016)", where such a calibration has been advocated for numerical schemes using global spectral analysis (GSA) for the convection equation. In recent times, due to growing computing power, simulations have been reported using pseudo-spectral methods, with spatial discretization performed in F...
The single dielectric barrier discharge (SDBD) plasma actuator has been developed in the present ... more The single dielectric barrier discharge (SDBD) plasma actuator has been developed in the present work for high-accuracy, high-performance computing of flow control applications. The present physics-based SDBD model is a significant improvement over the one developed by Bagade et al., [“Frequency-dependent capacitance–based plasma model for direct simulation of Navier–Stokes equation,” AIAA J. 55, 180–194 (2017)], which was used for planar geometry using sequential computation. Based on the physics of SDBD operation, phase-averaged fully developed body force over an ac cycle is computed and stored, which is reused. Thus, the intensive body force computations are bypassed in the new model, and the body force due to the SDBD plasma actuator is incorporated in the compressible Navier–Stokes equation that is solved in a body-fitted curvilinear coordinates. Here, the modified SDBD model enables performing large-scale simulations for the aerodynamic flow control at low speed and transonic ...
In this work, we have derived an optimized upwind compact difference scheme for achieving excelle... more In this work, we have derived an optimized upwind compact difference scheme for achieving excellent spatial resolution. The derived numerical scheme adds numerical diffusion which is strictly restricted to a high wavenumber region in order to control numerical instabilities, as well as to achieve de-aliasing ability. More importantly, the derived numerical scheme has excellent dispersion relation preserving (DRP) property. The applicability of the proposed scheme in simulating real flow problems has been demonstrated by solving flow inside a lid driven cavity, a transitional flow past AG24 aerofoil and a two-dimensional decaying turbulence flow.
Investigations have been performed via implicit large eddy simulations to study the overall effec... more Investigations have been performed via implicit large eddy simulations to study the overall effects of exciting a flow field by thermal (wall-heating and wall-cooling) and vortical (with high and low frequencies) actuation. The actuator is placed on the suction surface of a natural laminar flow (SHM-1) airfoil having an angle of attack of $\alpha = 0.38^o$ (cruise setting). Oncoming flow has a Mach number of $0.72$, and a Reynolds number based on chord of $Re = 16.2 \times 10^6$, for which a complex shock system is formed on the suction surface. Vorticity dynamics of the flow is studied using time series of vorticity at different locations above the suction surface and instantaneous contour plots of vorticity in the domain. An inspection of the flow using snapshots of $\nabla \rho $ and $\nabla (\rho T)$ is done to characterize the numerical Schlieren. Comparative effects of the various forms of excitation on the shock-boundary layer interactions (SBLI), have been analyzed using tim...
In Chap. 5, we have discussed the dynamics of the STWF for 2D transition. We have also shown the ... more In Chap. 5, we have discussed the dynamics of the STWF for 2D transition. We have also shown the inadequacy of the linear spatial instability studies in determining the evolution of disturbances. For monochromatic wall-excitation, the spatio-temporal evolution of disturbance was noted to depend on various factors like (a) excitation frequency, (b) amplitude, (c) exciter location and its width and (d) nature of excitation onset. In the present chapter, we would discuss about the 3D evolution of disturbances and the associated process of transition to turbulence. We first start with the governing equations, followed by numerical methods, problem definition and a brief description of boundary conditions. We have chosen the velocity-vorticity formulation of the incompressible NSE for its inherent accuracy to compute the 3D excitation of a nominally 2D ZPG boundary layer. Growth and evolution of disturbances, nature of vortical structures in the transitional and turbulent zones, and integral properties of the turbulent boundary layer (in terms of displacement and momentum thickness, shape factor and skin friction coefficient) are described subsequently.
Morkovin (Transition to turbulence, ASME FED Publication, USA, vol 114, pp 1–12, 1991, [33]) clas... more Morkovin (Transition to turbulence, ASME FED Publication, USA, vol 114, pp 1–12, 1991, [33]) classified transition to turbulence in to two main types: (i) The classical primary instability route whose onset is marked along with the presence of TS waves (as in ZPGBL) and (ii) the bypass routes, which encompass all other possible transition scenarios that do not exhibit TS waves. Unfortunately, this is too simplistic a classification scheme for the reasons given in the introduction. Moreover, the central theme of this chapter, is to show some typical bypass transition events shown experimentally and the corresponding theoretical explanations of these events. Of special interest is the development of two nonlinear theories of receptivity, derived from Navier–Stokes equation, without making any assumptions.
Mathematical Modeling and Computational Tools, 2020
We study the causal relation in a fluid dynamical system, for the impulse and frequency response ... more We study the causal relation in a fluid dynamical system, for the impulse and frequency response approaches as instability theories and corresponding experiments. The zero-pressure-gradient (ZPG) boundary layer is analyzed to find complementary aspects of these approaches. The drawbacks of instability study are in formulating it as a homogeneous system. Another difficulty for the instability is in classifying it for either temporal or spatial growth. When viscous effects were included in the spatial theory, it predicted wave solution (known as Tollmien–Schlichting (TS) waves), which left many scientists unconvinced. Experimental verification remained difficult as instability does not require explicit excitation, and dependence on background noise makes experiment non-repeatable. The classic experiment of Schubauer and Skramstad for the boundary layer (J Aero Sci 14(2), 69–78, [24]) excited a monochromatic source inside to obtain spatially growing TS waves—considered as the frequency response of the boundary layer. In contrast, Gaster and Grant (Proc R Soc A 347(1649), 253–269, [13]) tried to create TS waves by a localized impulse excitation and ended up creating a wave-packet by the impulse response of the dynamical system. Here, we focus mainly on the impulse response of the ZPG boundary layer using Bromwich contour integral method (BCIM) developed by the authors for spatio-temporal growth of disturbance field in creating spatio-temporal wave-front (STWF). The main achievement of BCIM is in identifying the cause for the creation of STWF by both the approaches.
Mathematical Modeling and Computational Tools, 2020
We study the causal relation in a fluid dynamical system, for the impulse and frequency response ... more We study the causal relation in a fluid dynamical system, for the impulse and frequency response approaches as instability theories and corresponding experiments. The zero-pressure-gradient (ZPG) boundary layer is analyzed to find complementary aspects of these approaches. The drawbacks of instability study are in formulating it as a homogeneous system. Another difficulty for the instability is in classifying it for either temporal or spatial growth. When viscous effects were included in the spatial theory, it predicted wave solution (known as Tollmien–Schlichting (TS) waves), which left many scientists unconvinced. Experimental verification remained difficult as instability does not require explicit excitation, and dependence on background noise makes experiment non-repeatable. The classic experiment of Schubauer and Skramstad for the boundary layer (J Aero Sci 14(2), 69–78, [24]) excited a monochromatic source inside to obtain spatially growing TS waves—considered as the frequency response of the boundary layer. In contrast, Gaster and Grant (Proc R Soc A 347(1649), 253–269, [13]) tried to create TS waves by a localized impulse excitation and ended up creating a wave-packet by the impulse response of the dynamical system. Here, we focus mainly on the impulse response of the ZPG boundary layer using Bromwich contour integral method (BCIM) developed by the authors for spatio-temporal growth of disturbance field in creating spatio-temporal wave-front (STWF). The main achievement of BCIM is in identifying the cause for the creation of STWF by both the approaches.
Here, the fundamental problem of Rayleigh–Taylor instability (RTI) is studied by direct numerical... more Here, the fundamental problem of Rayleigh–Taylor instability (RTI) is studied by direct numerical simulation (DNS), where the two air masses at different temperatures, kept apart initially by a non-conducting horizontal interface in a 2D box, are allowed to mix. Upon removal of the partition, mixing is controlled by RTI, apart from mutual mass, momentum, and energy transfer. To accentuate the instability, the top chamber is filled with the heavier (lower temperature) air, which rests atop the chamber containing lighter air. The partition is positioned initially at mid-height of the box. As the fluid dynamical system considered is completely isolated from outside, the DNS results obtained without using Boussinesq approximation will enable one to study non-equilibrium thermodynamics of a finite reservoir undergoing strong irreversible processes. The barrier is removed impulsively, triggering baroclinic instability by non-alignment of density, and pressure gradient by ambient disturbances via the sharp discontinuity at the interface. Adopted DNS method has dispersion relation preservation properties with neutral stability and does not require any external initial perturbations. The complete inhomogeneous problem with non-periodic, no-slip boundary conditions is studied by solving compressible Navier–Stokes equation, without the Boussinesq approximation. This is important as the temperature difference between the two air masses considered is high enough ($$\Delta T = 70$$ΔT=70 K) to invalidate Boussinesq approximation. We discuss non-equilibrium thermodynamical aspects of RTI with the help of numerical results for density, vorticity, entropy, energy, and enstrophy.
Rayleigh-Taylor instability (RTI) occurs when the interface between two fluids of different densi... more Rayleigh-Taylor instability (RTI) occurs when the interface between two fluids of different densities is removed, with the heavier (cold) fluid resting on top of the lighter (hot) fluid in the equilibrium state. This arrangement is unstable due to buoyancy, in the absence of any other forces. RTI is noted across a range of length scales from very small in nuclear fusion to supernova explosion at astrophysical scales. RTI is viewed as a baroclinic instability if viscous actions are ignored. An accurate non-overlapping parallel algorithm is used to solve a three-dimensional RTI problem, employing more than four billion points and a refined time step ($7.69 \times 10^{-8}{\rm s}$) for the direct numerical simulation. Air masses at two different temperatures are initially separated by a non-conducting partition inside a box (with a temperature difference of 200K). The impermeable partition is removed impulsively at $t =0$, and the ensuing instability is triggered by an acoustic mechanis...
The spatial stability properties of a mixed convection boundary layer developed over a heated hor... more The spatial stability properties of a mixed convection boundary layer developed over a heated horizontal flat plate is investigated under the linear and quasi-parallel assumptions by solving the Orr-Sommerfeld equations. The main aim of this work is to find a critical buoyancy parameter above which a qualitative change in the stability characteristics of the mixed convection boundary layer due to the influence of heat transfer is displayed. The equilibrium flow used here is that given by the similarity solution of [1] which implies the wall temperature to vary as the inverse square root of the distance from the leading edge of the plate. The Orr-Sommerfeld equation has been solved by using the compound matrix method (CMM)-- which locates all the modes in a chosen range of complex wave number, for spatial stability analysis. Cases with different buoyancy parameters have been investigated which were earlier reported incorrectly in [2]. Results indicate modifications of the hydrodynami...
Abstract We have investigated the dynamics of a shielded vortex kept in stagnant fluid, in front ... more Abstract We have investigated the dynamics of a shielded vortex kept in stagnant fluid, in front of plane wall at different heights. This placement instantaneously creates an unsteady boundary layer over the wall and the free vortex starts translation and rotation. Present study focuses not only on the dynamics of the free vortex and the underlying unsteady boundary layer, but a major aim is to explain various instabilities suffered by these two elements. Unlike other works reported on this topic, present computations did not encounter any singularities, which were identified as the harbinger of transition to turbulence. Present simulations are performed over significantly long times, due to the use of high accuracy dispersion relation preserving scheme and multi-dimensional filter.
Receptivity of a Bickley jet to a time harmonic symmetric (S- class) and anti-symmetric (AS- clas... more Receptivity of a Bickley jet to a time harmonic symmetric (S- class) and anti-symmetric (AS- class) vortical excitation is reported. Unlike wall bounded flows, the eigen-spectrum of jets reveal the presence of multiple dominant modes. The S- class displays the presence of upstream propagating disturbances. It is reasoned that due to limited streamwise extent of the domain, experiments and computations on round jets do not always correlate with the linear stability properties. For DNS, a new compact scheme (OUCS4), introduced in [6], along with RK4 time stepping is used. A new filtering procedure is advocated in the radial direction, which removes the numerical instability at the core (due to a mathematical singularity) and allows us to study receptivity of round jets to different classes of excitation.
Events in turbulent flows computed by direct numerical simulation (DNS) are often calibrated with... more Events in turbulent flows computed by direct numerical simulation (DNS) are often calibrated with properties based on homogeneous isotropic turbulence, advanced by Kolmogorov, and given in Turbulence, U. Frisch, Cambridge Univ. Press, UK (1995). However, these computational procedures are not calibrated using numerical analyses in order to assess their strengths and weaknesses for DNS. This is with the exception in "A critical assessment of simulations for transitional and turbulence flows- Sengupta, T.K., In Proc. of IUTAM Symp. on Advances in Computation, Modeling and Control of Transitional and Turbulent Flows, 491-532, (eds. Sengupta, Lele, Sreenivasan, Davidson), WSPC, Singapore (2016)", where such a calibration has been advocated for numerical schemes using global spectral analysis (GSA) for the convection equation. In recent times, due to growing computing power, simulations have been reported using pseudo-spectral methods, with spatial discretization performed in F...
The single dielectric barrier discharge (SDBD) plasma actuator has been developed in the present ... more The single dielectric barrier discharge (SDBD) plasma actuator has been developed in the present work for high-accuracy, high-performance computing of flow control applications. The present physics-based SDBD model is a significant improvement over the one developed by Bagade et al., [“Frequency-dependent capacitance–based plasma model for direct simulation of Navier–Stokes equation,” AIAA J. 55, 180–194 (2017)], which was used for planar geometry using sequential computation. Based on the physics of SDBD operation, phase-averaged fully developed body force over an ac cycle is computed and stored, which is reused. Thus, the intensive body force computations are bypassed in the new model, and the body force due to the SDBD plasma actuator is incorporated in the compressible Navier–Stokes equation that is solved in a body-fitted curvilinear coordinates. Here, the modified SDBD model enables performing large-scale simulations for the aerodynamic flow control at low speed and transonic ...
In this work, we have derived an optimized upwind compact difference scheme for achieving excelle... more In this work, we have derived an optimized upwind compact difference scheme for achieving excellent spatial resolution. The derived numerical scheme adds numerical diffusion which is strictly restricted to a high wavenumber region in order to control numerical instabilities, as well as to achieve de-aliasing ability. More importantly, the derived numerical scheme has excellent dispersion relation preserving (DRP) property. The applicability of the proposed scheme in simulating real flow problems has been demonstrated by solving flow inside a lid driven cavity, a transitional flow past AG24 aerofoil and a two-dimensional decaying turbulence flow.
Investigations have been performed via implicit large eddy simulations to study the overall effec... more Investigations have been performed via implicit large eddy simulations to study the overall effects of exciting a flow field by thermal (wall-heating and wall-cooling) and vortical (with high and low frequencies) actuation. The actuator is placed on the suction surface of a natural laminar flow (SHM-1) airfoil having an angle of attack of $\alpha = 0.38^o$ (cruise setting). Oncoming flow has a Mach number of $0.72$, and a Reynolds number based on chord of $Re = 16.2 \times 10^6$, for which a complex shock system is formed on the suction surface. Vorticity dynamics of the flow is studied using time series of vorticity at different locations above the suction surface and instantaneous contour plots of vorticity in the domain. An inspection of the flow using snapshots of $\nabla \rho $ and $\nabla (\rho T)$ is done to characterize the numerical Schlieren. Comparative effects of the various forms of excitation on the shock-boundary layer interactions (SBLI), have been analyzed using tim...
In Chap. 5, we have discussed the dynamics of the STWF for 2D transition. We have also shown the ... more In Chap. 5, we have discussed the dynamics of the STWF for 2D transition. We have also shown the inadequacy of the linear spatial instability studies in determining the evolution of disturbances. For monochromatic wall-excitation, the spatio-temporal evolution of disturbance was noted to depend on various factors like (a) excitation frequency, (b) amplitude, (c) exciter location and its width and (d) nature of excitation onset. In the present chapter, we would discuss about the 3D evolution of disturbances and the associated process of transition to turbulence. We first start with the governing equations, followed by numerical methods, problem definition and a brief description of boundary conditions. We have chosen the velocity-vorticity formulation of the incompressible NSE for its inherent accuracy to compute the 3D excitation of a nominally 2D ZPG boundary layer. Growth and evolution of disturbances, nature of vortical structures in the transitional and turbulent zones, and integral properties of the turbulent boundary layer (in terms of displacement and momentum thickness, shape factor and skin friction coefficient) are described subsequently.
Morkovin (Transition to turbulence, ASME FED Publication, USA, vol 114, pp 1–12, 1991, [33]) clas... more Morkovin (Transition to turbulence, ASME FED Publication, USA, vol 114, pp 1–12, 1991, [33]) classified transition to turbulence in to two main types: (i) The classical primary instability route whose onset is marked along with the presence of TS waves (as in ZPGBL) and (ii) the bypass routes, which encompass all other possible transition scenarios that do not exhibit TS waves. Unfortunately, this is too simplistic a classification scheme for the reasons given in the introduction. Moreover, the central theme of this chapter, is to show some typical bypass transition events shown experimentally and the corresponding theoretical explanations of these events. Of special interest is the development of two nonlinear theories of receptivity, derived from Navier–Stokes equation, without making any assumptions.
Mathematical Modeling and Computational Tools, 2020
We study the causal relation in a fluid dynamical system, for the impulse and frequency response ... more We study the causal relation in a fluid dynamical system, for the impulse and frequency response approaches as instability theories and corresponding experiments. The zero-pressure-gradient (ZPG) boundary layer is analyzed to find complementary aspects of these approaches. The drawbacks of instability study are in formulating it as a homogeneous system. Another difficulty for the instability is in classifying it for either temporal or spatial growth. When viscous effects were included in the spatial theory, it predicted wave solution (known as Tollmien–Schlichting (TS) waves), which left many scientists unconvinced. Experimental verification remained difficult as instability does not require explicit excitation, and dependence on background noise makes experiment non-repeatable. The classic experiment of Schubauer and Skramstad for the boundary layer (J Aero Sci 14(2), 69–78, [24]) excited a monochromatic source inside to obtain spatially growing TS waves—considered as the frequency response of the boundary layer. In contrast, Gaster and Grant (Proc R Soc A 347(1649), 253–269, [13]) tried to create TS waves by a localized impulse excitation and ended up creating a wave-packet by the impulse response of the dynamical system. Here, we focus mainly on the impulse response of the ZPG boundary layer using Bromwich contour integral method (BCIM) developed by the authors for spatio-temporal growth of disturbance field in creating spatio-temporal wave-front (STWF). The main achievement of BCIM is in identifying the cause for the creation of STWF by both the approaches.
Mathematical Modeling and Computational Tools, 2020
We study the causal relation in a fluid dynamical system, for the impulse and frequency response ... more We study the causal relation in a fluid dynamical system, for the impulse and frequency response approaches as instability theories and corresponding experiments. The zero-pressure-gradient (ZPG) boundary layer is analyzed to find complementary aspects of these approaches. The drawbacks of instability study are in formulating it as a homogeneous system. Another difficulty for the instability is in classifying it for either temporal or spatial growth. When viscous effects were included in the spatial theory, it predicted wave solution (known as Tollmien–Schlichting (TS) waves), which left many scientists unconvinced. Experimental verification remained difficult as instability does not require explicit excitation, and dependence on background noise makes experiment non-repeatable. The classic experiment of Schubauer and Skramstad for the boundary layer (J Aero Sci 14(2), 69–78, [24]) excited a monochromatic source inside to obtain spatially growing TS waves—considered as the frequency response of the boundary layer. In contrast, Gaster and Grant (Proc R Soc A 347(1649), 253–269, [13]) tried to create TS waves by a localized impulse excitation and ended up creating a wave-packet by the impulse response of the dynamical system. Here, we focus mainly on the impulse response of the ZPG boundary layer using Bromwich contour integral method (BCIM) developed by the authors for spatio-temporal growth of disturbance field in creating spatio-temporal wave-front (STWF). The main achievement of BCIM is in identifying the cause for the creation of STWF by both the approaches.
Here, the fundamental problem of Rayleigh–Taylor instability (RTI) is studied by direct numerical... more Here, the fundamental problem of Rayleigh–Taylor instability (RTI) is studied by direct numerical simulation (DNS), where the two air masses at different temperatures, kept apart initially by a non-conducting horizontal interface in a 2D box, are allowed to mix. Upon removal of the partition, mixing is controlled by RTI, apart from mutual mass, momentum, and energy transfer. To accentuate the instability, the top chamber is filled with the heavier (lower temperature) air, which rests atop the chamber containing lighter air. The partition is positioned initially at mid-height of the box. As the fluid dynamical system considered is completely isolated from outside, the DNS results obtained without using Boussinesq approximation will enable one to study non-equilibrium thermodynamics of a finite reservoir undergoing strong irreversible processes. The barrier is removed impulsively, triggering baroclinic instability by non-alignment of density, and pressure gradient by ambient disturbances via the sharp discontinuity at the interface. Adopted DNS method has dispersion relation preservation properties with neutral stability and does not require any external initial perturbations. The complete inhomogeneous problem with non-periodic, no-slip boundary conditions is studied by solving compressible Navier–Stokes equation, without the Boussinesq approximation. This is important as the temperature difference between the two air masses considered is high enough ($$\Delta T = 70$$ΔT=70 K) to invalidate Boussinesq approximation. We discuss non-equilibrium thermodynamical aspects of RTI with the help of numerical results for density, vorticity, entropy, energy, and enstrophy.
Rayleigh-Taylor instability (RTI) occurs when the interface between two fluids of different densi... more Rayleigh-Taylor instability (RTI) occurs when the interface between two fluids of different densities is removed, with the heavier (cold) fluid resting on top of the lighter (hot) fluid in the equilibrium state. This arrangement is unstable due to buoyancy, in the absence of any other forces. RTI is noted across a range of length scales from very small in nuclear fusion to supernova explosion at astrophysical scales. RTI is viewed as a baroclinic instability if viscous actions are ignored. An accurate non-overlapping parallel algorithm is used to solve a three-dimensional RTI problem, employing more than four billion points and a refined time step ($7.69 \times 10^{-8}{\rm s}$) for the direct numerical simulation. Air masses at two different temperatures are initially separated by a non-conducting partition inside a box (with a temperature difference of 200K). The impermeable partition is removed impulsively at $t =0$, and the ensuing instability is triggered by an acoustic mechanis...
The spatial stability properties of a mixed convection boundary layer developed over a heated hor... more The spatial stability properties of a mixed convection boundary layer developed over a heated horizontal flat plate is investigated under the linear and quasi-parallel assumptions by solving the Orr-Sommerfeld equations. The main aim of this work is to find a critical buoyancy parameter above which a qualitative change in the stability characteristics of the mixed convection boundary layer due to the influence of heat transfer is displayed. The equilibrium flow used here is that given by the similarity solution of [1] which implies the wall temperature to vary as the inverse square root of the distance from the leading edge of the plate. The Orr-Sommerfeld equation has been solved by using the compound matrix method (CMM)-- which locates all the modes in a chosen range of complex wave number, for spatial stability analysis. Cases with different buoyancy parameters have been investigated which were earlier reported incorrectly in [2]. Results indicate modifications of the hydrodynami...
Abstract We have investigated the dynamics of a shielded vortex kept in stagnant fluid, in front ... more Abstract We have investigated the dynamics of a shielded vortex kept in stagnant fluid, in front of plane wall at different heights. This placement instantaneously creates an unsteady boundary layer over the wall and the free vortex starts translation and rotation. Present study focuses not only on the dynamics of the free vortex and the underlying unsteady boundary layer, but a major aim is to explain various instabilities suffered by these two elements. Unlike other works reported on this topic, present computations did not encounter any singularities, which were identified as the harbinger of transition to turbulence. Present simulations are performed over significantly long times, due to the use of high accuracy dispersion relation preserving scheme and multi-dimensional filter.
Receptivity of a Bickley jet to a time harmonic symmetric (S- class) and anti-symmetric (AS- clas... more Receptivity of a Bickley jet to a time harmonic symmetric (S- class) and anti-symmetric (AS- class) vortical excitation is reported. Unlike wall bounded flows, the eigen-spectrum of jets reveal the presence of multiple dominant modes. The S- class displays the presence of upstream propagating disturbances. It is reasoned that due to limited streamwise extent of the domain, experiments and computations on round jets do not always correlate with the linear stability properties. For DNS, a new compact scheme (OUCS4), introduced in [6], along with RK4 time stepping is used. A new filtering procedure is advocated in the radial direction, which removes the numerical instability at the core (due to a mathematical singularity) and allows us to study receptivity of round jets to different classes of excitation.
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Papers by T. Sengupta