Numerical Methods for Solving Some Hydrodynamic Stability Problems
Tóm tắt
In this paper we solve the stability problem of the thermal convection in a thin fluid layer with free-free, slip-slip, and fixed-slip boundary conditions. We use different numerical methods to check their flexibility and accuracy where we use the following numerical methods: finite difference, High order finite difference,
$$p$$
order finite element, Chebyshev collocation-1, Chebyshev collocation-2 and Chebyshev tau. Moreover, we solve the equations of Poiseuille flow in a Brinkman porous media with slip-slip boundary conditions using
$$p$$
order finite element, Chebyshev collocation-1 and Chebyshev collocation-2 methods. The advantage and disadvantage of each method have been discussed. According to our result, we believe that the finite difference and finite element methods are very flexible methods and we can apply them to solve any problem easily. However, the accuracy of the the finite difference and finite element methods is less than the accuracy of the Chebyshev methods.
Tài liệu tham khảo
Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability. Dover, New York (1981)
Chang, M.H., Chen, F., Straughan, B.: Instability of Poiseuille flow in a fluid overlying a porous layer. J. Fluid Mech. 564, 287–303 (2006)
Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1981)
Dongarra, J., Straughan, B., Walker, D.: Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems. Appl. Numer. Math. 22, 399–435 (1996)
Fox, L.: Chebyshev methods for ordinary differential equations. Computer J. 4, 318–331 (1962)
Galdi, G.P., Robertson, A.M.: The relation between flow rate and axial pressure gradient for time-periodic Poiseuille flow in a pipe. J. Math. Fluid Mech. 7, S211–S223 (2005)
Galdi, G.P., Pileckas, K., Silvestre, A.L.: On the unsteady Poiseuille flow in a pipe. ZAMP 58, 994–1007 (2007)
Gheorghiu, C.I.: Spectral Methods for Non-Standard Eigenvalue Problems. Briefs in Mathematics. Springer, Berlin (2014)
Gheorghiu, C.I., Dragomirescu, F.: Spectral methods in linear stability. Applications to thermal convection with variable gravity field. Appl. Numer. Math. 59, 1290–1302 (2009)
Gheorghiu, C.I., Pop, I.S.: A modified Chebyshev-tau method for a hydrodynamic stability problem. Proceedings of ICAOR ’97, vol. II, pp. 119–126. Transylvania Press, Cluj-Napoca (1997)
Greenberg, L., Marletta, M.: The Ekman flow and related problems: spectral theory and numerical analysis. Math. Proc. Camb. Phil. Soc. 136, 719–764 (2004)
Harfash, A.J.: Magnetic effect on instability and nonlinear stability of double diffusive convection in a reacting fluid. Continuum Mech. Thermodyn. 25, 89–106 (2013)
Harfash, A.J.: Three dimensions simulation for the problem of a layer of non-Boussinesq fluid heated internally with prescribed heat flux on the lower boundary and constant temperature upper surface. Int. J. Eng. Sci. 74, 91–102 (2014a)
Harfash, A.J.: Three-dimensional simulations for convection in a porous medium with internal heat source and variable gravity effects. Transp. Porous Media 101, 281–297 (2014c)
Harfash, A.J.: Three dimensional simulation of radiation induced convection. Appl. Math. Comput. 227, 92–101 (2014d)
Harfash, A.J.: Three-dimensional simulations for convection problem in anisotropic porous media with nonhomogeneous porosity, thermal diffusivity, and variable gravity effects. Transp. Porous Media 102, 43–57 (2014e)
Harfash, A.J.: Three dimensional simulations for penetrative convection in a porous medium with internal heat sources. Acta Mech. Sin. 30, 144–152 (2014f)
Harfash, A.J.: Convection in a porous medium with variable gravity field and magnetic field effects. Transp. Porous Media (2014g). doi:10.1007/s11242-014-0305-8
Harfash, A.J.: Stability analysis of penetrative convection in anisotropic porous media with variable permeability. J. Non-Equilib. Thermodyn. (2014). doi:10.1515/jnet-2014-0009
Harfash, A.J., Hill, A.A.: Simulation of three dimensional double-diffusive throughflow in internally heated anisotropic porous media. Int. J. Heat Mass Trans. 72, 609–615 (2014)
Harfash, A.J., Straughan, B.: Magnetic effect on instability and nonlinear stability in a reacting fluid. Meccanica 47, 1849–1857 (2012)
Hill, A.A., Straughan, B.: A Legendre spectral element method for eigenvalues in hydromagnetic stability. J. Comput. Appl. Math. 193, 363–381 (2003)
Hill, A.A., Straughan, B.: Poiseuille flow of a fluid overlying a porous layer. J. Fluid Mech. 603, 137–149 (2008)
Hill, A.A., Straughan, B.: Stability of Poiseuille flow in a porous medium. In: Rannacher, R., Sequeira, A. (eds.) Advances in Mathematical Fluid Mechanics, pp. 287–293. Springer, Heidelberg (2010)
Kaiser, R., Mulone, G.: A note on nonlinear stability of plane parallel shear flows. J. Math. Anal. Appl. 302, 543–556 (2005)
Joseph, D.D.: Stability of Fluid Motions I. Springer, New York (1976)
Joseph, D.D.: On the stability of the Boussinesq equations. Arch. Rat. Mech. Anal. 20, 59–71 (1965)
Joseph, D.D.: Nonlinear stability of the Boussinesq equations by the method of energy. Arch. Ration. Mech. Anal. 22, 163–184 (1966)
Navier, C.L.M.H.: Mémoire sur les lois du mouvement des fluides. Mémoires de l’Academie Royale des Sciences d l’Institut de France 6, 389–440 (1823)
Ng, B.S., Reid, W.H.: An initial-value method for eigenvalue problems using compound matrices. J. Comput. Phys. 30, 125–136 (1979)
Ng, B.S., Reid, W.H.: On the numerical solution of the Orr-Sommerfeld problem: asymptotic initial conditions for shooting methods. J. Comput. Phys. 38, 275–293 (1980)
Ng, B.S., Reid, W.H.: The compound matrix method for ordinary differential systems. J. Comput. Phys. 58, 209–228 (1985)
Nield, D.A.: The stability of flow in a channel or duct occupied by a porous medium. Int. J. Heat Mass Transf. 46, 4351–4354 (2003)
Nield, D.A., Bejan, A.: Convection in porous media, 4th edn. Springer, New York (2013)
Orszag, S.: Accurate solutions of Orr-Sommerfeld stability equation. J. Fluid Mech. 50, 689–703 (1971)
Rayleigh, L.: On convective currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag. 32, 529–546 (1916)
Spotz, W.F: High-order compact finite difference schemes for computational mechanics. PHD thesis, University of Texas at Austin, Austin, Tx (1995)
Straughan, B.: Explosive Instabilities in Mechanics. Springer, Heidelberg (1998)
Straughan, B.: Stability, and Wave Motion in Porous Media, Volume 165 of Appl. Math. Sci. Springer, New York (2008)
Straughan, B., Harfash, A.J.: Instability in Poiseuille flow in a porous medium with slip boundary conditions. Microfluid. Nanofluid. 15, 109–115 (2013)
Straughan, B., Walker, D.W.: Two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems. J. Comput. Phys. 127, 128–141 (1996)
Webber, M.: The destabilizing effect of boundary slip on Bénard convection. Math. Methods Appl. Sci. 29, 819–838 (2006)
Webber, M., Straughan, B.: Stability of pressure driven flow in a microchannel. Rend. Circolo Matem. Palermo 29, 343–357 (2006)
Yecko, P.: Disturbance growth in two-fluid channel flow. The role of capillarity. Int. J. Multiph. Flow 34, 272–282 (2008)