Abstract
In this chapter, linear stability and receptivity analysis of the zero-pressure gradient (ZPG) boundary layer, under the parallel flow assumption, is discussed. This assumption implies that the equilibrium flow quantities do not grow in the streamwise direction and requires solving the Orr-Sommerfeld equation (OSE) to study evolution of disturbance field in a linearized analysis. The concept of the spatio-temporal wave-front (STWF) originates from the receptivity analysis with the OSE solved for the response field. First, the simplified description of equilibrium flow in terms of a similarity solution for ZPG boundary layer is presented. Following which the OSE is derived for boundary layers, making use of the parallel flow approximation (Drazin and Reid, Hydrodynamic stability, Cambridge University Press, UK, 1981, [19], Sengupta, Instabilities of flows and transition to turbulence, CRC Press, Taylor & Francis Group, Florida, USA, 2012, [53]). This equation have been solved for the ZPG boundary layer using analytical approaches in Heisenberg (Annalen der Physik Leipzig, 379:577–627, 1924, [28]), Schlichting (Nach Gesell d Wiss z G\(\ddot{\mathrm{o}}\)tt., MPK 42:181–208, 1933, [48]), Tollmien (NACA TM 609, 1931, [71]). We instead introduce the compound matrix method, a robust method for stiff differential equation useful for the OSE. Finally, the receptivity analysis of the ZPG boundary layer flow is provided, with results taken from Sengupta et al. (Phys Rev Lett, 96(22):224504, 2006, [61]), Sengupta et al. (Phys Fluids, 18:094101, 2006, [62]). The unique feature of the materials in this chapter is the topic of instability of mixed convection flows for which two theorems are enunciated for an inviscid linear mechanism, based on materials extensively taken from Sengupta et al., Physics of Fluids, 25, 094102 (2013).
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Notes
- 1.
[Reproduced from Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanism. Sengupta et al., Physics of Fluids, 25, 094102 (2013), with the permission of AIP Publishing.]
References
Allen, L., & Bridges, T. J. (2002). Numerical exterior algebra and the compound matrix method. Numerische Mathematik, 92, 197–232.
Alizard, F., & Robinet, J. (2007). Spatially convective global modes in a boundary layer. Physics of Fluids, 19, 114105.
Arfken, G. (1985). Mathematical methods for physicists (3rd ed.). Orlando: Academic Press.
Barkley, D., Gomes, M. G. M., & Henderson, R. D. (2002). Three-dimensional instability in flow over a backward-facing step. Journal of Fluid Mechanics, 473, 167–190.
Barkley, D., Blackburn, H. M., & Sherwin, S. J. (2008). Direct optimal growth analysis for timesteppers. International Journal for Numerical Methods in Fluids, 57, 1435–1458.
Bender, C. M., & Orszag, S. A. (1987). Advanced mathematical methods for scientists and engineers. Singapore: McGraw Hill Book Co., International Edition.
Bhaumik, S. (2013). Direct numerical simulation of inhomogeneous transitional and turbulent flows. Ph. D. thesis, I. I. T. Kanpur
Bhumkar, Y. G. (2011). High performance computing of bypass transition. Ph.D. thesis, I. I. T. Kanpur
Bhaumik, S., & Sengupta, T. K. (2014). Precursor of transition to turbulence: Spatiotemporal wave front. Physical Review E, 89(4), 043018.
Bhaumik, S., & Sengupta, T. K. (2017). Impulse response and spatio-temporal wave-packets: The common feature of rogue waves, tsunami and transition to turbulence. Physics of Fluids, 29, 124103.
Blackburn, H. M., Barkley, D., & Sherwin, S. J. (2008). Convective instability and transient growth in flow over a backward-facing step. Journal of Fluid Mechanics, 603, 271–304.
Brandt, L., & Henningson, D. S. (2002). Transition of streamwise streaks in zero-pressure-gradient boundary layers. Journal of Fluid Mechanics, 472, 229–261.
Brewstar, R. A., & Gebhart, B. (1991). Instability and disturbance amplification in a mixed-convection boundary layer. Journal of Fluid Mechanics, 229, 115–133.
Cebeci, T., & Bradshaw, P. (1977). Momentum transfer in boundary layers. Washington, DC: Hemisphere Publishing Corporation.
Chen, T. S., & Moutsoglu, A. (1979). Wave instability of mixed convection flow on inclined surfaces. Numerical Heat Transfer, 2, 497–509.
Chen, T. S., & Mucoglu, A. (1979). Wave instability of mixed convection flow over a horizontal flat plate. International Journal of Heat and Mass Transfer, 22, 185–196.
Chen, T. S., Sparrow, E. M., & Mucoglu, A. (1977). Mixed convection in boundary layer flow on a horizontal plate. ASME Journal of Heat Transfer, 99, 66–71.
Chomaz, J. M. (2005). Global instabilities in spatially developing flows: Non-normality and nonlinearity. Annual Review of Fluid Mechanics, 37, 357–392.
Drazin, P. G., & Reid, W. H. (1981). Hydrodynamic stability. UK: Cambridge University Press.
Eckert, E. R. G. & Soehngen, E. (1951). Interferometric studies on the stability and transition to turbulence in a free-convection boundary-layer. Proceedings of the General Discussion on Heat Transfer, ASME and IME London (Vol. 321) (1951)
Eiseman, P. R. (1985). Grid generation for fluid mechanics computation. Annual Review of Fluid Mechanics, 17, 487–522.
Fasel, H., & Konzelmann, U. (1990). Non-parallel stability of a flat-plate boundary layer using the complete Navier-Stokes equations. Journal of Fluid Mechanics, 221, 311–347.
Gaster, M. (1974). On the effect of boundary-layer growth on flow stability. Journal of Fluid Mechanics, 66(3), 465–480.
Gebhart, B., Jaluria, Y., Mahajan, R. L., & Sammakia, B. (1988). Buoyancy-induced flows and transport. Washington, DC: Hemisphere Publications.
Gilpin, R. R., Imura, H., & Cheng, K. C. (1978). Experiments on the onset of longitudinal vortices in horizontal Blasius flow heated from below. ASME Journal of Heat Transfer, 100, 71–77.
Haaland, S. E., & Sparrow, E. M. (1973). Vortex instability of natural convection flows on inclined surfaces. International Journal of Heat Mass Transfer, 16, 2355–2367.
Hall, P., & Morris, H. (1992). On the instability of boundary layers on heated flat plates. Journal of Fluid Mechanics, 245, 367–400.
Heisenberg, W. (1924). \(\ddot{\rm U}\)ber stabilit\({\ddot{\rm a}}\)t und turbulenz von fl\({\ddot{\rm u}}\)ssigkeitsstr\({\ddot{\rm o}}\)men. Annalen der Physik Leipzig, 379, 577–627 (Translated as ‘On stability and turbulence of fluid flows’. NACA Tech. Memo. Wash. No 1291 1951)
Iyer, P. A., & Kelly, R. E. (1974). The instability of the laminar free convection flow induced by a heated, inclined plate. International Journal of Hear Mass Transfer, 17, 517–525.
Jain, M. K., Iyengar, S. R. K. & Jain, R. K. (2003). Numerical methods for scientific and engineering computation. New Delhi: New Age International
Kaiktsis, L., Karniadakis, G. M., & Orszag, S. A. (1991). Onset of three-dimensionality, equibria and early transition in flow over a backward-facing step. Journal of Fluid Mechanics, 231, 501–528.
Kaiktsis, L., Karniadakis, G. M., & Orszag, S. A. (1996). Unsteadiness and convective instabilities in two-dimensional flow over a backward-facing step. Journal of Fluid Mechanics, 321, 157–187.
Kloker, M., Konzelmann, U., & Fasel, H. (1993). Outflow boundary conditions for spatial Navier-Stokes simulations of transitional boundary layers. AIAA Journal, 31, 620.
Kreyszig, E. (1999). Advanced engineering mathematics. Singapore: Wiley.
Leal, L. G. (1973). Steady separated flow in a linearly decelerated free stream. Journal of Fluid Mechanics, 59, 513–535.
Liu, Z., & Liu, C. (1994). Fourth order finite difference and multigrid methods for modeling instabilities in flat plate boundary layer-2D and 3D approaches. Computers and Fluids, 23, 955–982.
Lloyd, J. R., & Sparrow, E. M. (1970). On the instability of natural convection flow on inclined plates. Journal of Fluid Mechanics, 42, 465–470.
Lord, R. (1880). On the stability or instability of certain fluid motions. Scientific Papers, 1, 361–371.
Marquet, O., Sipp, D., Chomaz, J. M., & Jacquin, L. (2008). Amplifier and resonator dynamics of a low Reynolds-number recirculation bubble in a global framework. Journal of Fluid Mechanics, 605, 429–443.
Moutsoglu, A., Chen, T. S., & Cheng, K. C. (1981). Vortex instability of mixed convection flow over a horizontal flat plate. ASME Journal of Heat Transfer, 103, 257–261.
Mucoglu, A., & Chen, T. S. (1978). Wave instability of mixed convection flow along a vertical flat plate. Numerical Heat Transfer, 1, 267–283.
Mureithi, E. W., & Denier, J. P. (2010). Absolute-convective instability of mixed forced-free convection boundary layers. Fluid Dynamics Research, 372, 517–534.
Ng, B. S., & Reid, W. H. (1980). On the numerical solution of the Orr-Sommerfeld problem: Asymptotic initial conditions for shooting method. Journal of Computational Physics, 38, 275–293.
Ng, B. S., & Reid, W. H. (1985). The compound matrix method for ordinary differential systems. Journal of Computational Physics, 58, 209–228.
Rajpoot, M. K., Sengupta, T. K., & Dutt, P. K. (2010). Optimal time advancing dispersion relation preserving schemes. Journal of Computational Physics, 229(10), 3623–3651.
Saric, W. S., & Nayfeh, A. H. (1975). Nonparallel stability of boundary-layer flows. Physics of Fluids, 18(8), 945–950.
Schlatter, P., & \(\ddot{O}rl\ddot{u}\), R. (2012). Turbulent boundary layers at moderate Reynolds numbers. Journal of Fluid Mechanics, 710, 5–34.
Schlichting, H. (1933). Zur entstehung der turbulenz bei der plattenstr\({\ddot{\rm o}}\)mung. Nach. Gesell. d. Wiss. z. G\({\ddot{\rm o}}\)tt., MPK,42, 181–208
Schneider, W. (1979). A similarity solution for combined forced and free convection flow over a horizontal plate. International Journal of Heat and Mass Transfer, 22, 1401–1406.
Schubauer, G. B., & Skramstad, H. K. (1947). Laminar boundary layer oscillations and the stability of laminar flow. Journal of Aerosol Science, 14(2), 69–78.
Sengupta T. K. (1990). Receptivity of a growing boundary layer to surface excitation. (Unpublished manuscript).
Sengupta, T. K. (1991). Impulse response of laminar boundary layer and receptivity. In C. Taylor (Ed.), Proceedings of the 7th International Conference Numerical Methods in Laminar and Turbulent Layers. Stanford University
Sengupta, T. K. (2012). Instabilities of flows and transition to turbulence. Florida, USA: CRC Press, Taylor & Francis Group.
Sengupta, T. K. (2013). High accuracy computing methods: Fluid flows and wave phenomenon. USA: Cambridge University Press.
Sengupta, T. K., & Bhaumik, S. (2011). Onset of turbulence from the receptivity stage of fluid flows. Physical Review Letters, 154501, 1–5.
Sengupta, T. K., & Venkatasubbaiah, K. (2006). Spatial stability for mixed convection boundary layer over a heated horizontal plate. Studies in Applied Mathematics, 117, 265–298.
Sengupta, T. K., Ballav, M., & Nijhawan, S. (1994). Generation of Tollmien-Schlichting waves by harmonic excitation. Physics of Fluids, 6(3), 1213–1222.
Sengupta, T. K., Bhaumik, S., Singh, V., & Shukl, S. (2009). Nonlinear and nonparallel receptivity of zero-pressure gradient boundary layer. International Journal of Emerging Multidisciplinary Fluid Sciences, 1, 19–35.
Sengupta, T. K., Chattopadhyay, M., Wang, Z. Y., & Yeo, K. S. (2002). By-pass mechanism of transition to turbulence. Journal of Fluids and Structures, 16, 15–29.
Sengupta, T. K., De, S., & Sarkar, S. (2003). Vortex-induced instability of an incompressible wall-bounded shear layer. Journal of Fluid Mechanics, 493, 277–286.
Sengupta, T. K., Rao, A. K., & Venkatasubbaiah, K. (2006). Spatiotemporal growing wave fronts in spatially stable boundary layers. Physical Review Letters, 96(22), 224504.
Sengupta, T. K., Rao, A. K., & Venkatasubbaiah, K. (2006). Spatiotemporal growth of disturbances in a boundary layer and energy based receptivity analysis. Physics of Fluids, 18, 094101.
Sengupta, T. K., Sircar, S. K., & Dipankar, A. (2006). High accuracy compact schemes for DNS and acoustics. Journal of Scientific Computing, 26(2), 151–193.
Sengupta, T. K., Unnikrishnnan, S., Bhaumik, S., Singh, P., & Usman, S. (2011). Linear spatial stability analysis of mixed convection boundary layer over a heated plate. Program in Applied Mathematics, 1(1), 71–89.
Sengupta, T. K., Bhaumik, S., & Bhumkar, Y. (2012). Direct numerical simulation of two-dimensional wall-bounded turbulent flows from receptivity stage. Physical Review E, 85(2), 026308.
Sengupta, T. K., Bhaumik, S., & Bose, R. (2013). Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanisms. Physics of Fluids, 25, 094102.
Shaukatullah, H., & Gebhart, B. (1978). An experimental investigation of natural convection flow on an inclined surface. International Journal of Heat and Mass Transfer, 21, 1481–1490.
Sparrow, E. M., & Husar, R. B. (1969). Longitudinal vortices in natural convection flow on inclined plates. Journal of Fluid Mechanics, 37, 251–255.
Sparrow, E. M., & Minkowycz, W. J. (1962). Buoyancy effects on horizontal boundary-layer flow and heat transfer. International Journal of Heat and Mass Transfer, 5, 505–511.
Skote, M., Haritonidis, J. H., & Henningson, D. S. (2002). Varicose instabilities in turbulent boundary layers. Physics of Fluids, 14, 2309–2323.
Tollmien, W. (1931). \(\ddot{\rm U}\)ber die enstehung der turbulenz. I, English translation. NACA TM 609
Tumin, A. (2003). The spatial stability of natural convection flow on inclined plates. ASME Journal of Fluids Engineering, 125, 428–437.
Unnikrishnan, S. (2011). Linear stability analysis and nonlinear receptivity study of mixed convection boundary layer developing over a heated flat plate. M. Tech. thesis (I.I.T. Kanpur, 2011)
Van der Pol, B., & Bremmer, H. (1959). Operational calculus based on two-sided Laplace integral. Cambridge, UK: Cambridge University Press.
Wang, X. A. (1982). An experimental study of mixed, forced, and free convection heat transfer from a horizontal flat plate to air. ASME Journal of Heat Transfer, 104, 139–144.
Wu, R. S., & Cheng, K. C. (1976). Thermal instability of Blasius flow along horizontal plates. International Journal of Heat and Mass Transfer, 19, 907–913.
Zhang, S. L. (1997). GPBi-CG: Generalized product-type methods based on Bi-CG for solving Non symmetric linear systems. SIAM Journal on Scientific Computing, 18(2), 537–551.
Zuercher, E. J., Jacobs, J. W., & Chen, C. F. (1998). Experimental study of the stability of boundary-layer flow along a heated inclined plate. J. Fluid Mech., 367, 1–25.
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Sengupta, T.K., Bhaumik, S. (2019). Receptivity and Instability. In: DNS of Wall-Bounded Turbulent Flows. Springer, Singapore. https://doi.org/10.1007/978-981-13-0038-7_3
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