Papers by Swagata Bhaumik
Dispersion analysis of numerical schemes using 2D compressible linearized Navier–Stokes equation for direct numerical simulation
Computers & Fluids
Compound Matrix Method for Linear Stability Analysis of Flow Over a Flat Plate
Lecture notes in mechanical engineering, 2024
arXiv (Cornell University), Jan 30, 2024
We investigate the linear stability of parallel two-dimensional (2D) compressible boundary layer ... more We investigate the linear stability of parallel two-dimensional (2D) compressible boundary layer flow over a smooth, adiabatic flat plate. We consider both two-and three-dimensional (3D) disturbances, which yield 6 th -and 8 th -order systems for wall-normal variation of spectral amplitudes, respectively. We perform spatial stability analysis of the flow using the compound matrix method (CMM) to remove the stiffness of the problem, unlike the conventional techniques using Gram-Schmidt ortho-normalization or discretizing the governing equations by appropriate finite difference schemes. This method has yet to be employed for linear stability analysis of compressible boundary layers. We consider the Mach number M of the flow to vary from low subsonic cases (M = 0.1) to supersonic cases (M = 6). First, we validate the compound matrix method by comparing the M = 0.1 case results with the incompressible boundary layer governed by the Orr-Sommerfeld equation. The results depict an excellent match for primary and secondary modes. The flow becomes increasingly stable with an increase in M up to M = 2. We further note subsonic cases to be more stable when considering 3D disturbances. On the contrary, this conclusion holds only for 3D disturbances having spanwise wavenumber β greater than a specific optimum value corresponding to supersonic cases up to M = 2. We show that 2D disturbances are more spatially unstable than 3D perturbations with no spanwise variation (i.e., β = 0) up to M = 1.4, and the opposite scenario happens for 1.4 < M < 2. Only one primary spatially unstable mode is found for boundary layer cases up to M = 2. We locate a series of unstable modes for M > 3, and the number of such modes is much more than two, as reported in Mack (AGARD Report No. 709, Part 3, 1984) from viscous calculations. Mack reports only two unstable modes up to M = 4.5 that subsequently fuse with an increase in M . Our results show the number and the frequency extent of the unstable zone for these modes increase significantly with an increase in Mach number, e.g., we find 6, 8, and 10 spatially unstable modes up to streamwise wavenumber of α r = 0.4 for M = 3, 4 and 6, respectively. The calculation of group velocity for these modes shows that these propagate downstream at a higher speed of 0.78 to 0.89 times the free-stream speed than those corresponding to incompressible, subsonic, and low supersonic (M < 2) cases. While the shape of the neutral curves for the second unstable
Nonlinear Theoretical and Computational Analysis of Fluid Flows
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.
A High Accuracy Preserving Parallel Algorithm for Compact Schemes for DNS
ACM Transactions on Parallel Computing, Oct 16, 2020
A new accuracy-preserving parallel algorithm employing compact schemes is presented for direct nu... more A new accuracy-preserving parallel algorithm employing compact schemes is presented for direct numerical simulation of the Navier-Stokes equations. Here the connotation of accuracy preservation is having the same level of accuracy obtained by the proposed parallel compact scheme, as the sequential code with the same compact scheme. Additional loss of accuracy in parallel compact schemes arises due to necessary boundary closures at sub-domain boundaries. An attempt to circumvent this has been done in the past by the use of Schwarz domain decomposition and compact filters in “A new compact scheme for parallel computing using domain decomposition,” J. Comput. Phys. 220, 2 (2007), 654--677, where a large number of overlap points was necessary to reduce error. A parallel compact scheme with staggered grids has been used to report direct numerical simulation of transition and turbulence by the Schwarz domain decomposition method. In the present research, we propose a new parallel algorithm with two benefits. First, the number of overlap points is reduced to a single common boundary point between any two neighboring sub-domains, thereby saving the number of points used, with resultant speed-up. Second, with a proper design, errors arising due to sub-domain boundary closure schemes are reduced to a user designed error tolerance, bringing the new parallel scheme on par with sequential computing. Error reduction is achieved by using global spectral analysis, introduced in “Analysis of central and upwind compact schemes,” J. Comput. Phys. 192, 2, (2003) 677--694, which analyzes any discrete computing method in the full domain integrally. The design of the parallel compact scheme is explained, followed by a demonstration of the accuracy of the method by solving benchmark flows: (1) periodic two-dimensional Taylor-Green vortex problem; (2) flow inside two-dimensional square lid-driven cavity (LDC) at high Reynolds number; and (3) flow inside a non-periodic three-dimensional cubic LDC with the staggered grid arrangement.
Physics of Fluids, May 1, 2018
In the present research, a new theory of instability based on enstrophy is presented for incompre... more In the present research, a new theory of instability based on enstrophy is presented for incompressible flows. Explaining instability through enstrophy is counter-intuitive, as it has been usually associated with dissipation for the Navier-Stokes equation (NSE). This developed theory is valid for both linear and nonlinear stages of disturbance growth. A previously developed nonlinear theory of incompressible flow instability based on total mechanical energy described in the work of Sengupta et al. ["Vortex-induced instability of an incompressible wall-bounded shear layer," J. Fluid Mech. 493, 277-286 (2003)] is used to compare with the present enstrophy based theory. The developed equations for disturbance enstrophy and disturbance mechanical energy are derived from NSE without any simplifying assumptions, as compared to other classical linear/nonlinear theories. The theory is tested for bypass transition caused by free stream convecting vortex over a zero pressure gradient boundary layer. We explain the creation of smaller scales in the flow by a cascade of enstrophy, which creates rotationality, in general inhomogeneous flows. Linear and nonlinear versions of the theory help explain the vortex-induced instability problem under consideration.
3D Routes of Transition to Turbulence by STWF
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.
Physical Review E, Feb 8, 2012
Deterministic route to turbulence creation in 2D wall boundary layer is shown here by solving ful... more Deterministic route to turbulence creation in 2D wall boundary layer is shown here by solving full Navier-Stokes equation by dispersion relation preserving (DRP) numerical methods for flow over a flat plate excited by wall and free stream excitations. Present results show the transition caused by wall excitation is predominantly due to nonlinear growth of the spatiotemporal wave front, even in the presence of Tollmien-Schlichting (TS) waves. The existence and linear mechanism of creating the spatiotemporal wave front was established in Sengupta, Rao and Venkatasubbaiah [Phys. Rev. Lett. 96, 224504 (2006)] via the solution of Orr-Sommerfeld equation. Effects of spatiotemporal front(s) in the nonlinear phase of disturbance evolution have been documented by Sengupta and Bhaumik [Phys. Rev. Lett. 107, 154501 (2011)], where a flow is taken from the receptivity stage to the fully developed 2D turbulent state exhibiting a k -3 energy spectrum by solving the Navier-Stokes equation without any artifice. The details of this mechanism are presented here for the first time, along with another problem of forced excitation of the boundary layer by convecting free stream vortices. Thus, the excitations considered here are for a zero pressure gradient (ZPG) boundary layer by (i) monochromatic time-harmonic wall excitation and (ii) free stream excitation by convecting train of vortices at a constant height. The latter case demonstrates neither monochromatic TS wave, nor the spatiotemporal wave front, yet both the cases eventually show the presence of k -3 energy spectrum, which has been shown experimentally for atmospheric dynamics in Nastrom, Gage and Jasperson [Nature 310, 36 (1984)]. Transition by a nonlinear mechanism of the Navier-Stokes equation leading to k -3 energy spectrum in the inertial subrange is the typical characteristic feature of all 2D turbulent flows. Reproduction of the spectrum noted in atmospheric data (showing dominance of the k -3 spectrum over the k -5/3 spectrum in Nastrom et al.) in laboratory scale indicates universality of this spectrum for all 2D turbulent flows. Creation of universal features of 2D turbulence by a deterministic route has been established here for the first time by solving the Navier-Stokes equation without any modeling, as has been reported earlier in the literature by other researchers.
Journal of Computational Physics, Mar 1, 2015
Accuracy of velocity-vorticity ( V, ω)-formulations over other formulations in solving Navier-Sto... more Accuracy of velocity-vorticity ( V, ω)-formulations over other formulations in solving Navier-Stokes equation has been established in recent times. However, the issue of non-satisfaction of solenoidality conditions on vorticity is not addressed in the literature which can possibly lead to non-physical solution. In this respect, here, we have developed and reported conservative rotational form of the ( V, ω)-formulation which preserves the solenoidality condition on vorticity in a much simpler way compared to other formulations. Superiority of rotational form over the conventional Laplacian form of ( V, ω)-formulation is also shown [by comparing the results for flows inside cubical lid driven cavity (LDC)]. For solving the 3D Navier-Stokes equation using a staggered grid, we use optimized compact schemes for (a) interpolation and (b) evaluation of first and second derivatives. As illustrations, we have solved problems of (i) flow inside a 3D lid driven cavity (LDC), whose solutions are compared with experimental results reported by Koseff and Street [J. Fluids Engg. 106, 390-398 (1984)] and (ii) 3D
Investigation of the Plume Dynamics and the Near-field of a Supersonic Twinjet
A new compact difference scheme for second derivative in non-uniform grid expressed in self-adjoint form
Journal of Computational Physics, Mar 1, 2011
... For computing unsteady, incompressible, viscous flows with two unbounded directions, authors ... more ... For computing unsteady, incompressible, viscous flows with two unbounded directions, authors in [20] obtained the potential velocity component by solving the Laplace s equation in ... In the following, we briefly describe the HOC scheme with respect to a second order ...
Investigation of a Twinjet Configuration with and without Flow Control
Jet noise has been an active area of research owing to public health concerns and aviation regula... more Jet noise has been an active area of research owing to public health concerns and aviation regulations. This problem is further amplified with the use of multiple engines placed in closed proximity to each other. In addition to significantly altered farfield noise as compared to a single-jet, previous experimental results show high levels of dynamic pressure fluctuations in the inter-nozzle region which can cause structural damage. In this work, Large Eddy Simulations are performed to study the dynamics of a supersonic twinjet configuration, with and without flow control. Results indicate that relative to single-jet configuration, twinjets show decrease in column lengths due to significant modifications in shear-layer properties. The simulations also corroborate previous experimental observations of very high levels of dynamic pressure fluctuations in the inter-nozzle region relative to a single-jet configuration for identical flow parameters. The phenomenon of noise shielding along the plane containing the two jets has been reported for all the twinjet cases explored in this work and trends of sound pressure levels obtained at various polar angles indicate an increased efficiency of shielding at lower polar angles. The sensitivity of the dynamic pressure fluctuations on the nozzle-exit boundary layer thickness is also characterized. Simulations are also performed to mimic the effects of Localized Arc Filament Plasma Actuators at different excitation frequencies (St=0.3 and 0.9) actuated in an axisymmetric fashion (m=0). Close to the nozzle exit, toroidal structures formed as a result of control result in significant increase in jet spreading leading to the formation of a recirculation region along the symmetry plane, relative to the uncontrolled case. This excitation scheme increased the levels of dynamic pressure fluctuations closer to the nozzle exit in the inter-nozzle region. In contrast, the near field shows significant reduction in pressure fluctuations, especially for the higher Strouhal number case.
Nonlinear Theoretical and Computational Analysis of Fluid Flows
DNS of Wall-Bounded Turbulent Flows, 2018
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.
A High Accuracy Preserving Parallel Algorithm for Compact Schemes for DNS
ACM Transactions on Parallel Computing, 2020
A new accuracy-preserving parallel algorithm employing compact schemes is presented for direct nu... more A new accuracy-preserving parallel algorithm employing compact schemes is presented for direct numerical simulation of the Navier-Stokes equations. Here the connotation of accuracy preservation is having the same level of accuracy obtained by the proposed parallel compact scheme, as the sequential code with the same compact scheme. Additional loss of accuracy in parallel compact schemes arises due to necessary boundary closures at sub-domain boundaries. An attempt to circumvent this has been done in the past by the use of Schwarz domain decomposition and compact filters in “A new compact scheme for parallel computing using domain decomposition,” J. Comput. Phys. 220, 2 (2007), 654--677, where a large number of overlap points was necessary to reduce error. A parallel compact scheme with staggered grids has been used to report direct numerical simulation of transition and turbulence by the Schwarz domain decomposition method. In the present research, we propose a new parallel algorith...
Physical Review E, 2019
The investigation on grid sensitivity for the bifurcation problem of the canonical lid-driven cav... more The investigation on grid sensitivity for the bifurcation problem of the canonical lid-driven cavity (LDC) flow results is reported here with very fine grids. This is motivated by different researchers presenting different first bifurcation critical Reynolds number (Re cr1 ), which appears to depend on the formulation, numerical method, and choice of grid. By using a very-high-accuracy parallel algorithm, and the same method with which sequential results were presented by Lestandi et al. [Comput. Fluids 166, 86 (2018)] [for (257 × 257) and (513 × 513) uniformly spaced grid], we present results using (1025×1025) and (2049×2049) grid points. Detailed results presented using these grids help us understand the computational physics of the numerical receptivity of the LDC flow, with and without explicit excitation. The mathematical physics of the investigated problem will become apparent when we identify the roles of numerical errors with the ambient omnipresent disturbances in real physical flows as interchangeable. In physical or in numerical setups, presence of disturbances cannot be ignored. In this context, the need for explicit excitation for the used compact scheme arises for a definitive threshold amplitude, below which the flow relaxes back to quiescent state after the excitation is removed in computations. We also implement the present parallel method to show the physical aspects of primary and secondary instabilities to be maintained for other numerical schemes, and we show the results to reflect the complex physics during multiple subcritical Hopf bifurcation. Also, we relate the various sources of errors during computations that is typical of such shear-driven flow. These results, with near spectral accuracy, constitute universal benchmark results for the solution of Navier-Stokes equation for LDC.
Computers & Fluids, 2018
Two-dimensional (2D) flow inside a lid driven cavity (LDC) is shown to display multi-modal behavi... more Two-dimensional (2D) flow inside a lid driven cavity (LDC) is shown to display multi-modal behavior in a consistent manner following the first Hopf bifurcation with varying Reynolds numbers (Re), depending upon the chosen spatial and temporal discretization scheme. Direct numerical simulation (DNS) following impulsive start, is used to show spatio-temporal growth and its nonlinear saturation of disturbance growth. Despite the fact that researchers have produced different value of Reynolds number when first Hopf bifurcation occurs (Re cr1 ), DNS fundamentally differs from classical bifurcation studies involving global instability study of an equilibrium flow due to adopted nonlinear approach and not restricting the analysis to temporal instability only. The accuracy attribute of the DNS adopted here has been shown conclusively earlier via demonstration of a weak transient polygonal core vortex surrounded by relatively stronger gyrating vortices, which appear as a constellation after the disappearance of the transient, in Sengupta et al. (J. Comput. Phys., 228, 3048-3071 and 6150-6168 (2009)). Investigated LDC flow is characterized by multiple time scales at any Re, which are weak function of Re in selective intervals, punctuated by multiple bifurcations. The present investigation achieves two primary goals. First, it proposes to reconcile that Re cr1 obtained by different numerical approaches can be shown to be in same range, provided the equilibrium flow obtained
Physics of Fluids, 2018
In the present research, a new theory of instability based on enstrophy is presented for incompre... more In the present research, a new theory of instability based on enstrophy is presented for incompressible flows. Explaining instability through enstrophy is counter-intuitive, as it has been usually associated with dissipation for the Navier-Stokes equation (NSE). This developed theory is valid for both linear and nonlinear stages of disturbance growth. A previously developed nonlinear theory of incompressible flow instability based on total mechanical energy described in the work of Sengupta et al. [“Vortex-induced instability of an incompressible wall-bounded shear layer,” J. Fluid Mech. 493, 277–286 (2003)] is used to compare with the present enstrophy based theory. The developed equations for disturbance enstrophy and disturbance mechanical energy are derived from NSE without any simplifying assumptions, as compared to other classical linear/nonlinear theories. The theory is tested for bypass transition caused by free stream convecting vortex over a zero pressure gradient boundary...
Investigation of the Plume Dynamics and the Near-field of a Supersonic Twinjet
46th AIAA Fluid Dynamics Conference, 2016
Physics of Twinjet Plume Interactions
54th AIAA Aerospace Sciences Meeting, 2016
Different Routes of Transition by Spatio-Temporal Wave-Front
Advances in Computation, Modeling and Control of Transitional and Turbulent Flows, 2015
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Papers by Swagata Bhaumik