Entropy Generation through the Interaction of Laminar Boundary-Layer Flows: Sensitivity to Initial Conditions
- 1 Department of Mechanical Engineering, University of Utah, Salt Lake City, UT, USA
Abstract
A modified form of the Townsend equations for the fluctuating velocity wave vectors is applied to the interaction of a longitudinal vortex with a laminar boundary-layer flow. These three-dimensional equations are cast into a Lorenz-format system of equations for the spectral velocity component solutions. Tsallis-form empirical entropic indices are obtained from the solutions of the modified Lorenz equations. These solutions are sensitive to the initial conditions applied to the time-dependent coupled, non-linear differential equations for the spectral velocity components. Eighteen sets of initial conditions for these solutions are examined. The empirical entropic indices yield corresponding intermittency exponents which then yield the entropy generation rates for each set of initial conditions. The flow environment consists of the flow of hydrogen gas with impurities at a given temperature and pressure in the interaction of a longitudinal vortex with a laminar boundary layer flow. Results are presented that indicate a strong correlation of predicted entropy generation rates and the corresponding applied initial conditions. These initial conditions may be ascribed to the turbulence levels in the boundary layer, thus indicating a source for the subsequent entropy generation rates by the interactive instabilities.
- Townsend, A.A. (1976) The Structure of Turbulent Shear Flow. 2nd Edition, Cambridge University Press, Cambridge.
- Sparrow, C. (1982) The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer-Verlag, New York. https://doi.org/10.1007/978-1-4612-5767-7
- Isaacson, L.K. (2017) Entropy, 19, 278. https://doi.org/10.3390/e19060278
- Isaacson, L.K. (2016) Entropy, 18, 47. https://doi.org/10.3390/e18020047
- Isaacson, L.K. (2016) Entropy, 18, 279. https://doi.org/10.3390/e18080279
- Walsh, E.J. and Hernon, E. (2006) Entropy, 8, 25-30. https://doi.org/10.3390/e8010025
- Hellberg, C.S. and Orszag, S.A. (1988) Physics of Fluids, 31, 6-8. https://doi.org/10.1063/1.867010
- Singer, B.A. (1996) Physics of Fluids, 8, 509-521. https://doi.org/10.1063/1.868804
- Ersoy, S. and Walker, J.D.A. (1985) Physics of Fluids, 28, 2687-2698. https://doi.org/10.1063/1.865226
- Belotserkovskii, O.M. and Khlopkov, Y.I. (2010) Monte Carlo Methods in Mechanics of Fluid and Gas. World Scientific Publishing, Singapore, 101-102.
- Schmid, P.J. and Henningson, D.S. (2001) Stability and Transition in Shear Flows. Springer-Verlag, New York, 401-465. https://doi.org/10.1007/978-1-4613-0185-1_9
- Cebeci, T. and Bradshaw, P. (1977) Momentum Transfer in Boundary Layers. Hemisphere, Washington DC, 319-321.
- Cebeci, T. and Cousteix, J. (2005) Modeling and Computation of Boundary-Layer Flows. Horizons Publishing, Long Beach.
- Hansen, A.G. (1964) Similarity Analyses of Boundary Value Problems in Engineering. Prentice-Hall, Englewood Cliffs, 86-92.
- Sengupta, T.K. (2012) Instabilities of Flows and Transition to Turbulence. CRC Press, Boca Raton, 171-200.
- Attard, P. (2012) Non-Equilibrium Thermodynamics and Statistical Mechanics: Foundation and Applications. Oxford University Press, Oxford. https://doi.org/10.1093/acprof:oso/9780199662760.001.0001
- Manneville, P. (1990) Dissipative Structures and Weak Turbulence. Academic Press, San Diego.
- Isaacson, L.K. (2012) Entropy, 14, 131-160. https://doi.org/10.3390/e14020131
- Pecora, L.M. and Carroll, T.L. (1996) Synchronization in Chaotic Systems. In: Kapitaniak, T., Ed., Controlling Chaos: Theoretical and Practical Methods in Nonlinear Dynamics, Academic Press Inc., San Diego, 142-145. https://doi.org/10.1016/B978-012396840-1/50040-0