Professor

TIFR Centre for Interdisciplinary Sciences

21, Brundavan Colony, Narsingi,

Hyderabad - 500075.

For Newtonian fluids in relatively simple geometries, laminar flow is well-studied both experimentally and theoretically. We know a lot about the statistical characteristics of homogeneous isotropic turbulence, but not so much about the turbulence in shear flows that occurs all around us. What we understand the least is what makes a certain laminar flow become turbulent, and also what is the route for completing the transition. Instability and transition to turbulence is a focus area of this group.

This group is also interested in looking into the physics of interfacial flows like the formation of hydraulic jumps and droplet evolution, where surface tension plays a major role. A description of ongoing exploratory work, in many of which we don't have answers yet, is given below.

- 1979-1984 : B.Tech. in Chemical Engineering, IIT Delhi.
- 1984-1986 : M.S. in Chemical Engineering, Drexel University, Philadelphia.
- 1991-1994 : Ph.D in Aerospace Engineering, IISc Bangalore.
- 1994 : Post-Doctoral Research, Dept. of Aeronautics, Caltech
- 1988-1998 : Scientist in the Computational and Theoretical Fluid Dynamics Division, National Aerospace Laboratories, Bangalore.
- 1998-2012: At the Jawaharlal Nehru Center for Advanced Scientific Research, in a faculty position
- Present : Professor, TCIS, TIFR

PhD Students

Mamta Raju Jotkar: (co-advised by Vassilis Theofilis, University of Madrid)

Mamta is working on global stability of flows through complex geomtries. She wishes to understand the route to turbulence in fluids. This picture shows a typical linear eigenmode she has found in a diverging channel. This is the amplitude of the disturbance streamfunction of a given frequency.

(co-advisor: R. Narasimha, JNCASR Bangalore)

Croor Singh works on the fluid mechanics of cloud flows. One part of his work so far uses toy-models with point vortices and inertial particles; the other involves studying the Rayleigh-Benard convection of mixtures of water vapour and air. Through these and other studies, he hopes to some day connect cloud microphysics and large-scale flows.

He reads books, blogs, news reports, slashdot entries, papers and articles about biology, language, economics, and politics. He writes a somewhat popular blog at http://croor.wordpress.com

Sharath Jose is trying to understand disturbance evolution in parallel shear flows with density and viscosity stratifications. He also intends to carry forward the analysis to the full nonlinear system using direct numerical simulations, in collaboration with Luca Brandt at KTH, Stockholm. A sample result is shown here given where the optimal stream function for 2D velocity perturbations at Re = 1000 is shown in the figures.

There are changes in the maximum growth, the optimal streamwise wave number and the time in which the maximum growth is obtained.

Divya Venkataraman is doing her PhD at the University of Genova, where her advisor is Alessandro Bottaro, and at the JNCASR.

She has interesting computational results on a "hairfoil", an airfoil with long cilia (hair) stuck on a part of it. She has shown that drag and lift are modified considerably by the cilia (due to modification in the vortex-shedding process), and is trying to explain the physics behind it.

For this, she is developing a reduced-order model that predicts the vortex-shedding patterns behind an airfoil coated with such a poro-elastic layer of bristles. Such a model, involving the structural parameters of the bristles, will consequently lead to finding parameters that can possibly enhance the aerodynamic performance.

Difference between time-averaged pressure fields close to prpelastically-coated and smooth airfaoil at 45 angel of attack. The positive-magnitude difference, particularly on the suction side, signifies a smaller negative magnitude of mean pressure for the coated case, yielding lower mean drag.

Is at the Raman Research Institute where his main advisor is Ranjini Bandyopadhyay. He is working on the dynamics of complex fluids like clay solutions.

Alumni (at the JNCASR)

PhD

Ã‚Â | Thesis topic | Year | Present Affiliation |

Sumesh P.T. | Statics and dynamics of drops on solid surfaces: theory and simulations | 2012 | Oxford University (joining soon as postDoc) |

Harish N Dixit | Some studies on vortices with density stratification | 2011 | University of British Columbia (postdoc) |

Ratul DasGupta | A computational and semi-analytical study of laminar, standing hydraulic jumps | 2011 | Weizmann Institute of Science, Israel. (postdoc) |

Gayathri Swaminathan | Global stability studies of some spatially developing flows | 2011 | Airbus Industrie Bangalore |

N. Vinod | Stability and transition in boundary layers : effect of transverse curvature and pressure gradient | 2007 | IIT Gandhinagar (on the faculty) |

Kirti Chandra Sahu | Novel stability problems in pipe flows | 2007 | IIT Hyderabad (on the faculty) |

A. Sameen | Stability of plane channel flow with viscosity stratification (co-advisor: Prof. ON Ramesh, Aero. Dept., IISc) | 2005 | IIT Madras, Chennai (on the faculty) |

Anubhab Roy | The effect of elasticity on stability of vortical flows (main advisor: G Subramanian) | 2012 | Cornell (postdoc) |

MS

Ã‚Â | Thesis topic | Year |

Rohith V. S. | Vortex interactions and merger | 2012 |

Srikanth T | Patterns in the flow past rectangular oscillating cylinders | 2011 |

Rahul Bale | Transient growth of instabilities in stratified channel flow | 2009 |

Kaushik Srinivasan | Direct numerical simulation of transition in unstably stratified Poiseuille flow | 2007 |

Pinaki | A linear stability analysis of the spatially-developing incompressible plane | Ã‚Â |

Bhattacharyya | laminar mixing layer using minimal composite theory (co-advisor: R. Narasimha) | 2007 |

Antina Ghosh | Simulation of nano-scale flows by molecular dynamics methods (co-advisor: Srikanth Sastry) | 2005 |

Kirti Chandra Sahu | Numerical computation of spatially developing flows by full multigrid technique. | 2003 |

For small density differences, the Boussinesq approximation is usually made. We showed that non-Boussinesq effects cannot be neglected in the vicinity of vortices if the density difference, however small, occurs over a short lateral extent. Our discovery of a new spiral Kelvin-Helmholtz instability appeared on the cover of the Journal of Fluid Mechanics in 2010, as shown. An initially flat density interface is seen to be rolled by a vortex into a Lituus spiral, before breaking down. Co-author, Harish Dixit.

A vortex under such circumstances is broken down rapidly, displaying decaying turbulence, but the Boussinesq approximation will predict a lifetime wrong by orders of magnitude. This has implications, for instance in the inverse cascade picture of two-dimensional turbulence. Other counter-intuitive instabilities can result from wave interactions in this scenario, with quasimodes acting as interfacial modes.

My group is also interested in the physics of interfacial flows, including the statics and dynamics of drops, and wetting and flow problems in the presence of gravity and Marangoni stresses. An example of this: The shallow water equations are considered valid when the ratio Ã¯ÂÂ¥ of the typical lengths normal and parallel to the wall is small. Along with a recent graduate, Ratul DasGupta, and Gaurav Tomar who visited us, I showed that the shallow water equations are incorrect at any order inÃ¯â‚¬Â Ã¯ÂÂ¥ whenever the Froude number is close to unity, i.e., when inertial effects from the streamwise momentum balance cancel out gravitational effects. Here non-hydrostatic contributions to the pressure become important. This has implications for film flows on inclined surfaces as well, which we are now investigating.

Hydraulic jumps are known to be caused due to the adverse pressure gradient created by gravity, but we found a gravity-free hydraulic jump. This picture is from a 2007 paper in Physical Review Letters, of a femtolitre capacity silver cup whose walls form a Ã¢â‚¬Å“nano-vadaÃ¢â‚¬Â, created by a droplet of molten metal impinging on a surface slightly cooler than its melting point. The experiment is from the group of GU Kulkarni. The hydraulic jump here is caused by surface tension due to the azimuthal curvature, acting like a rubber band tightening around the flow.

An important source of mixing in the ocean is due to double-diffusive instabilities, caused by density stratification, and the vastly different diffusion rates of the two `solutesÃ¢â‚¬â„¢, namely salt and temperature, causing this stratification. The closed curves on the left enclose a new double diffusive instability in shear flow, when density is constant throughout the fluid, but viscosity is not. This occurs when viscosity is `stablyÃ¢â‚¬â„¢ stratified. Work with Kirti Sahu, J. Fluid Mech. 2011.

- Inertial particles in leap-frogging vortex-pair flow. S. Ravichandran, Rama Govindarajan and Roddam Narasimha, IUTAM Symposium on mobile particulate systems.
- Spatio-temporal linear stability of double-diffusive two-fluid channel flow. Kirti Chandra Sahu & Rama Govindarajan. Physics of Fluids, 24, 054103.
- Oscillatory settling in wormlike-micelle solutions: bursts and a long time scale. Nitin Kumar, Sayantan Majumdar, Aditya Sood, Rama Govindarajan, Sriram Ramaswamy & A. K. Sood. Soft Matter, 8, 4310-4313.
- Secondary instabilities in incompressible axisymmetric boundary layers: effect of transverse curvature. N Vinod & Rama Govindarajan, Journal of Fluids Engineering, ASME. 134, 2, 024503.
- The hydraulic jump and the shallow water equations, Ratul Dasgupta and Rama Govindarajan, International Journal of Advances in Engineering Sciences and Applied Mathematics, Special Issue, 3 (1Ã¢â‚¬â€œ4):126Ã¢â‚¬â€œ130

- Linear stability of double diffusive two-fluid channel flow. Kirti Chandra Sahu & Rama Govindarajan. Journal of Fluid Mechanics, 687, 529-539.
- Minimum energy shapes of one-side-pinned static drops on inclined surfaces. Sumesh P.T. and Rama Govindarajan, Physical Review E, 84, 046304.
- Stability of a vortex in radial density stratification: role of wave interactions, Harish N Dixit & Rama Govindarajan, Journal of Fluid Mechanics, 679, 582 Ã¢â‚¬â€œ 615.
- Global instabilities in diverging channel flows, Gayathri Swaminathan, Kirti Chandra Sahu, A Sameen & Rama Govindarajan, Theoretical and Computational Fluid Dynamics, 25, 53-64.
- Vortex shedding patterns, their competition, and chaos in flow past inline oscillating rectangular cylinders, Srikanth T, Harish N Dixit, Rao Tatavarti & Rama Govindarajan, Physics of Fluids 23, 073603.
- Flow control via porous coating of compliant actuators: a simple interactive model Flow control via porous coating of compliant actuators: a simple interactive model, Divya Venkataraman, Alessandro Bottaro and Rama Govindarajan, Proc. IUTAM Symposium on Bluff Body Wakes.

- Non-similar solutions of the viscous shallow water equations governing weak hydraulic jumps. Ratul Dasgupta & Rama Govindarajan, Physics of Fluids 22, 112108.
- The laminar flow through a circular pipe is known to be linearly stable at any Reynolds number, but Kirti Sahu showed (J. Fluid Mech, 2005) that any non-zero divergence of the pipe will result in a finite critical Reynolds number, which can be quite low even for small divergences. There is a simple inviscid explanation for the power law obtained.
- Vortex-induced instabilities and accelerated collapse due to inertial eÃ¯Â¬â‚¬ects of density stratiÃ¯Â¬Âcation, Harish
- N Dixit & Rama Govindarajan, Journal of Fluid Mechanics 646, 2010, 415 -439. Cover picture article.
- The possible equilibrium shapes of static pendant drops, P T Sumesh & Rama Govindarajan, Journal of Chemical Physics 133, 144707.
- An introduction to hydrodynamic stability, Anubhab Roy & Rama Govindarajan, in Rheology of Complex Fluids ed. J Murali Krishnan, Springer.
- Transient growth and why we should care about it, Rahul Bale & Rama Govindarajan, Resonance, 15, No. 5, 441-457.
- Shedding behaviour in Ã¯Â¬â€šow past an inline oscillating square cylinder, Srikanth T., Harish N Dixit & Rama Govindarajan, Proc. IUTAM Symposium on Bluff Body Wakes.

- Instabilities due a vortex at a density interface: gravitational and centrifugal effects. Harish N Dixit & Rama Govindarajan, Proc. Seventh IUTAM Symposium on Laminar-Turbulent Transition, Ed. Schlatter & Henningson, Springer.
- Global Instabilities in Wall Jets. Gayathri Swaminathan, A Sameen & Rama Govindarajan, Proc. Seventh IUTAM Symposium on Laminar-Turbulent Transition, Ed. Schlatter & Henningson, Springer.
- Two-dimensional flow field in the vicinity of an accelerating bodies. Srikanth Rao, Rao Tatavarti & Rama Govindarajan. Proc. Second National Conference of the Ocean Society of India.
- Global instabilities in non-symmetric convergent-divergent channels. Gayathri Swaminathan & Rama Govindarajan, Proc. Symposium on Global Flow Instability and Control.

- The relative roles of divergence and velocity slip in the stability of plane channel flow, K.C. Sahu, A. Sameen and R. Govindarajan, European Physical Journal, Appl. Phys., 44, 101107.
- Random Vortex-Street Model for a Self-Similar Plane Turbulent Jet, Victor S. Lvov, Anna Pomyalov, Itamar Procaccia and Rama Govindarajan, Physical Review Letters, 101, 094503.
- In what sense is a low-Reynolds number mixing layer stable? R. Narasimha, Pinaki Bhattacharya and Rama Govindarajan, Proc. 12th Asian Congress of Fluid Mechanics, Daejeon, Korea.

- The effect of wall heating on instability of channel flow. A Sameen & Rama Govindarajan, Journal of Fluid Mechanics, 577, 417-442. Corrigendum in 673 (2011), 603-605.
- Stability of miscible core-annular flows with viscosity stratification. B. Selvam, S. Merk, Rama Govindarajan & E. Meiburg, Journal of Fluid Mechanics, 592, 23-49.
- Gravity-free hydraulic jumps and metal femtolitre cups. Manikandan Mathur, Ratul DasGupta, Neena Susan John, N. R. Selvi, G. U. Kulkarni, & Rama Govindarajan, Physical Review Letters, 98, 164502.
- Linear instability of entry flow in a pipe, Kirti Chandra Sahu & Rama Govindarajan, ASME Journal of Fluids Engineering, 129, Issue 10, 1277-1280.
- The effectiveness of wall heating as a control option for channel flow. A Sameen & Rama Govindarajan, Sadhana, 32, 65-81.
- The signature of laminar instabilities in the zone of transition to turbulence. N. Vinod & Rama Govindarajan, Journal of Turbulence, 8, 1-17.
- A general method for linear instability computations for flow through arbitrary two-dimensional geometries. Gayathri Swaminathan & Rama Govindarajan, Proc. Seventh Asian CFD Conference.
- Study of a lid-driven cavity in an axisymmetric geometry. Ratul DasGupta & Rama Govindarajan, Proc. IUTAM Symposium on Unsteady Separated Flows and their Control.

- The critical Reynolds number of a laminar incompressible mixing layer from minimal composite theory. Pinaki Bhattacharya, Maalika Manoharan, Rama Govindarajan & R Narasimha, Journal of Fluid Mechanics, 565, 105-114.
- A Facile Method of Producing Femtoliter Metal Cups by Pulsed Laser Ablation. Neena Susan John, N. R. Selvi, Manikandan Mathur, Rama Govindarajan & G. U. Kulkarni, J. Physical Chemistry B, 110, 22975-22978.
- Flow past a rectangular cylinder in a stratified fluid. Harish Dixit, A Sameen, Rama Govindarajan, Proc. Sixth Intl. Symposium on Stratified Flows, Perth, Australia (Ed: G.N. Ivey), 189-194.
- Direct numerical simulation of a pressure driven channel flow with stable thermal stratification. Kaushik Srinivasan & Rama Govindarajan, Proceedings of the 3rd BSME -ASME International Conference on Thermal Engineering, Dhaka.
- Instability of entry flow in a pipe. Kirti Chandra Sahu & Rama Govindarajan, Proc. Eleventh Asian Congress of Fluid Mechanics.
- Is a stable laminar mixing layer possible? Pinaki Bhattacharya, Rama Govindarajan & R Narasimha, Proc. Eleventh Asian Congress of Fluid Mechanics.

- Manipulating transition to turbulence: A viscosity stratification does not always help. Vijayakumar Chikkadi, A. Sameen & Rama Govindarajan, Physical Review Letters, 95, 264504.
- Dynamical systems model of entrainment due to coherent structures. Srevatsan Muralidharan, KR Sreenivas & Rama Govindarajan, Physical Review E, 72, 046308.
- Drag reduction by polymer additives in decaying turbulence. Chirag Kalelkar, Rama Govindarajan, & Rahul Pandit, Physical Review E, 72, 017301.
- Stability of flow through a slowly diverging pipe. Kirti Chandra Sahu & Rama Govindarajan, Journal of Fluid Mechanics, 531, 325-334.
- Accurate estimate of disturbance amplitude variation from solution of minimal composite stability theory. Rama Govindarajan and R. Narasimha, Theoretical and Computational Fluid Dynamics, 19 (4): 229-235.
- Instabilities and transition in boundary layers. N. Vinod and Rama Govindarajan, Pramana J. Physics, special issue, 64, No. 3, 323-332.
- Super-linear speedup of a parallel multigrid Navier-Stokes solver on Flosolver. T. N. Venkatesh, V. R. Sarasamma, Rajalakshmy S., Kirti C. Sahu & Rama Govindarajan, Current Science, 88, No. 4, 589-593.
**Article:**Tsunami relief: an observerÃ¢â‚¬â„¢s viewpoint. 2005. Proceedings of Six months after the Tsunami, NIAS, Bangalore.

- The pattern of breakdown of laminar flow into turbulent spots. N Vinod & Rama Govindarajan, Physical Review Letters, 93, 114501.
- Effect of miscibility on the linear instability of two-fluid channel flow. Rama Govindarajan, International Journal of Multiphase Flow, 30/10, 1177-1192.
- A minimal composite theory for stability of non-parallel compressible boundary-layer flow. Sanjeev Rao, Rajeswari Seshadri & Rama Govindarajan, Theoretical and Computational Fluid Dynamics, 17, No. 4, 233-248.
- The segregation instability of a sheared suspension film. Rama Govindarajan, PR Nott & S Ramaswamy, Proc. Tenth Asian Cong. Fluid Mech., B31.

- Shell model for drag reduction with polymer additive in homogeneous turbulence. Roberto Benzi, Elisabetta De Angelis, Rama Govindarajan & Itamar Procaccia, Phys. Rev. E, 68, 016308.
- Stabilization of hydrodynamic flows by small viscosity variations. Rama Govindarajan, Victor S. LÃ¢â‚¬â„¢vov, Itamar Procaccia & A. Sameen, Phys. Rev. E, 67, 026310.
- Stripes in sheared non-brownian suspensions with a free surface. Rama Govindarajan, P. R. Nott & S. Ramaswamy, Physica A, 318/1-2, 80-84.
- Are subtransitions in boundary layers a signature of laminar instability? N Vinod & Rama Govindarajan, Proc. International Symposium on Advances in Fluid Mech., 200-210.

- Ã¢â‚¬Å“Universal behaviour of entrainment due to coherent structures in turbulent shear flow,Ã¢â‚¬Â Rama Govindarajan, Phys. Rev. Lett., 88, 134503.
- Ã¢â‚¬Å“Stability analysis of an axisymmetric boundary layer,Ã¢â‚¬Â N. Vinod, Hamsa Balakrishnan & Rama Govindarajan, Proc. Ninth Asian Cong. Fluid Mech.
- Ã¢â‚¬Å“Secondary instabilities in stratified channel flow,Ã¢â‚¬Â A. Sameen, O.N. Ramesh & Rama Govindarajan, Proc. Ninth Asian Cong. Fluid Mech.
- Ã¢â‚¬Å“Instabilities in spatially developing flows: a fresh perspective,Ã¢â‚¬Â Rama Govindarajan, Proc. Ninth Asian Cong. Fluid Mech.
- Ã¢â‚¬Å“Surprising effects of minor viscosity gradients,Ã¢â‚¬Â Rama Govindarajan, J. Indian Inst. Sci., 82 121-127.

- Ã¢â‚¬Å“Retardation of the onset of turbulence by minor viscosity contrasts,Ã¢â‚¬Â Rama Govindarajan, V. S. LÃ¢â‚¬â„¢vov & I. Procaccia, Phys. Rev. Lett., 87, 174501.
- Ã¢â‚¬Å“Estimating amplitude ratios in boundary layer stability theory: a comparison between two approaches,Ã¢â‚¬Â Rama Govindarajan & R. Narasimha, J. Fluid Mech., 439 403-412.
- Ã¢â‚¬Å“Stabilization and destabilization of channel flow by location of viscosity-stratified fluid layer,Ã¢â‚¬Â Balaji T. Ranganathan & Rama Govindarajan, Phys. Fluids (Lett.), 13 No. 1, 1-3.
- Ã¢â‚¬Å“A theory of suspension segregation in horizontal couette cells,Ã¢â‚¬Â Rama Govindarajan, P. R. Nott & S. Ramaswamy, Phys. Fluids (Lett.) 13, 12, 3517-3520.
- Ã¢â‚¬Å“Stability of non-parallel flows: minimal composite theoriesÃ¢â‚¬Â, R. Narasimha & Rama Govindarajan, Proc. 14th Australasian Fluid Mech. Conf.
- Ã¢â‚¬Å“The transition zone in adverse pressure gradient boundary layers: results from stochastic simulations,Ã¢â‚¬Â N. Vinod & Rama Govindarajan, Proc. XV Intl. Symp. Air Breathing Engines, 1093.

- Ã¢â‚¬Å“Transition delay by surface heating: a zonal analysis for axisymmetric bodies,Ã¢â‚¬Â Rama Govindarajan & R. Narasimha, J. Fluid Mech., 418, 77-100.
- Ã¢â‚¬Å“The role of the critical layer in the stability of viscous shear flows,Ã¢â‚¬Â Rama Govindarajan, Current Sci., 79, 6, 741-746.
- Ã¢â‚¬Å“Minimal composite equations and the stability of non-parallel flows,Ã¢â‚¬Â R. Narasimha & Rama Govindarajan, Current Sci., 79, 6, 730-740.

- 1999 : Ã¢â‚¬Å“A low-order parabolic theory for two-dimensional boundary layer stability,Ã¢â‚¬Â Rama Govindarajan & R. Narasimha, Phys. Fluids, 11 No. 6, 1449-1458.
- 1999 : Ã¢â‚¬Å“Low-order Stability Theory for Compressible Boundary Layer Flow,Ã¢â‚¬Â Deepanjan Mitra, Rajeswari Seshadri, Sanjeev Rao & Rama Govindarajan, in Laminar-Turbulent Transition, Springer, 487-492.
- 1999 : The transition zone on a heated axisymmetric body," Rama Govindarajan & R. Narasimha, in Laminar-Turbulent Transition, Springer, 351-356.
- 1999 : Ã¢â‚¬Å“Stability of weakly non-similar swept-wing boundary layers,Ã¢â‚¬Â Rama Govindarajan & R. Narasimha, in Laminar-Turbulent Transition, Springer, 193-198.
- 1999 : Ã¢â‚¬Å“Turbulence and Flying Machines,Ã¢â‚¬Â Resonance, 4, No. 11, 54-62.
- 1998 : Ã¢â‚¬Å“Mixing and chaotic fluid particle trajectories in the flow around two leap-frogging vortex rings,Ã¢â‚¬Â Rama Govindarajan, A. Leonard & S. Wiggins, in Lecture Notes in Physics, ed. C.-H. Bruneau, Springer, 482-487.
- 1998 : Ã¢â‚¬Å“Jamming mechanisms in homogeneous and heterogeneous traffic flow,Ã¢â‚¬Â Rizwan Ameer, Deepak Raj M & Rama Govindarajan, Proc. 3rd Asian Comput. Fluid. Dyn. Conf., 2, 144-149.
- 1998 : Ã¢â‚¬Å“Stability of swept wing boundary layers: a low-order theory,Ã¢â‚¬Â Rama Govindarajan, R. Narasimha & Avinash Lohit, Proc. 3rd Asian Comput. Fluid. Dyn. Conf., 2, 56-61.
- 1998 : Ã¢â‚¬Å“Vortex sheets in plane strain and the fine structure of turbulence,Ã¢â‚¬Â R. Narasimha & Rama Govin-darajan, in Recent Advances in Fluid Mechanics, ed. by PL Sachdev & M Venkatachalappa, Gordon & Breach, 1 67-75.
- 1997 : Ã¢â‚¬Å“A low-order theory for stability of non-parallel boundary layer flows,Ã¢â‚¬Â Rama Govindarajan & R. Narasimha, Proc. Roy. Soc. Lond. A, 453, 2537-2549.
- 1997 : Ã¢â‚¬Å“Absolute instability in spatially developing two-dimensional wakes,Ã¢â‚¬Â Rama Govindarajan, R. Narasimha & J.S.B. Gajjar, Proc. 7th Asian Cong. Fluid Mech., 1, 163-166.
- 1995 : Ã¢â‚¬Å“Stability of spatially developing boundary layers in pressure gradients,Ã¢â‚¬Â Rama Govindarajan & R. Narasimha, J. Fluid Mech., 300, 117-147.
- 1991 : Ã¢â‚¬Å“The role of residual nonturbulent disturbances on transition onset in two-dimensional boundary layers,Ã¢â‚¬Â Rama Govindarajan & R. Narasimha, J. Fluids Engg., Trans. ASME, 113, 147-149.