Becker–Morduchow–Libby solution
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Becker–Morduchow–Libby solution is an exact solution of the compressible Navier–Stokes equations, that describes the structure of one-dimensional shock waves. The solution was discovered in a restrictive form by Richard Becker in 1922, which was generalized by Morris Morduchow and Paul A. Libby in 1949.[1][2] The solution was also discovered independently by M. Roy and L. H. Thomas in 1944[3][4]The solution showed that there is a non-monotonic variation of the entropy across the shock wave. Before these works, Lord Rayleigh obtained solutions in 1910 for fluids with viscosity but without heat conductivity and for fluids with heat conductivity but without viscosity.[5] Following this, in the same year G. I. Taylor solved the whole problem for weak shock waves by taking both viscosity and heat conductivity into account.[6][7]
Mathematical description
[edit]In a frame fixed with a planar shock wave, the shock wave is steady. In this frame, the steady Navier–Stokes equations for a viscous and heat conducting gas can be written as
where is the density, is the velocity, is the pressure, is the internal energy per unit mass, is the temperature, is an effective coefficient of viscosity, is the coefficient of viscosity, is the second viscosity and is the thermal conductivity. To this set of equations, one has to prescribe an equation of state and an expression for the energy in terms of any two thermodynamics variables, say . Instead of , it is convenient to work with the specific enthalpy
Let us denote properties pertaining upstream of the shock with the subscript "" and downstream with "". The shock wave speed itself is denoted by . The first integral of the governing equations, after imposing the condition that all gradients vanish upstream, are found to be
By evaluating these on the downstream side where all gradients vanish, one recovers the familiar Rankine–Hugoniot conditions, , and Further integration of the above equations require numerical computations, except in one special case where integration can be carried out analytically.
Analytical solution
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Becker%E2%80%93Morduchow%E2%80%93Libby_solution.jpg/300px-Becker%E2%80%93Morduchow%E2%80%93Libby_solution.jpg)
Two assumptions has to be made to facilitate explicit integration of the third equation. First, assume that the gas is ideal (polytropic since we shall assume constant values for the specific heats) in which case the equation of state is and further , where is the specific heat at constant pressure and is the specific heat ratio. The third equation then becomes
where is the Prandtl number based on ; when , say as in monoatomic gases, this Prandtl number is just the ordinary Prandtl number . The second assumption made is so that the terms inside the parenthesis becomes a total derivative, i.e., . This is a reasonably good approximation since in normal gases, Pradntl number is approximately equal to . With this approximation and integrating once more by imposing the condition that is bounded downstream, we find[8]
This above relation indicates that the quantity is conserved everywhere, not just on the upstream and downstream side. Since for the polytropic gas , where is the specific volume and is the sound speed, the above equation provides the relation between the ratio and the corresponding velocity (or density or specific volume) ratio
- ,
i.e.,[8]
where is the Mach number of the wave with respect to upstream and . Combining this with momentum and continuity integrals, we obtain the equation for as follows
We can introduce the reciprocal-viscosity-weighted coordinate[9]
where , so that
The equation clearly exhibits the translation invariant in the -direction which can be fixed, say, by fixing the origin to be the location where the intermediate value is reached. Using this last condition, the solution to this equation is found to be[8]
As (or, ), we have and as (or, ), we have This ends the search for the analytical solution. From here, other thermodynamics variables of interest can be evaluated. For instance, the temperature ratio is esaily to found to given by
and the specific entropy , by
The analytical solution is plotted in the figure for and . The notable feature is that the entropy does not monotonically increase across the shock wave, but it increases to a larger value and then decreases to a constant behind the shock wave. Such scenario is possible because of the heat conduction, as it will become apparent by looking at the entropy equation which is obtained from the original energy equation by substituting the thermodynamic relation , i.e.,
While the viscous dissipation associated with the term always increases the entropy, heat conduction increases the entropy in the colder layers where , whereas it decreases the entropy in the hotter layers where .
Taylor's solution: Weak shock waves
[edit]When , analytical solution is possible only in the weak shock-wave limit, as first shown by G. I. Taylor in 1910.[6] In the weak shock-wave limit, all terms such as , etc., will be small. The thickness of the shock wave is of the order so that differentiation with respect to increases the order smallness by one; e.g. is a second-order small quantity. Without going into the details and treating the gas to a generic gas (not just polytropic), the solution for is found to be related to the steady travelling-wave solution of the Burgers' equation and is given by[10]
where
in which is the Landau derivative (for polytropic gas ) and is a constant which when multiplied by some characteristic frequency squared provides the acoustic absorption coefficient.[11] The specific entropy is found to be proportional to and is given by
Note that is a second-order small quantity, although is a third-order small quantity as can be inferred from the above expression which shows that for both . This is allowed since , unlike , passes through a maximum within the shock wave.
Validity of continuum hypothesis: since the thermal velocity of the molecules is of the order and the kinematic viscosity is of the order , where is the mean free path of the gas molecules,, we have ; an estimation based on heat conduction gives the same result. Combining this with the relation , shows that
i.e., the shock-wave thickness is of the order the mean free path of the molecules. However, in the continuum hypothesis, the mean free path is taken to be zero. It follows that the continuum equations alone cannot be strictly used to describe the internal structure of strong shock waves; in weak shock waves, can be made as small as possible to make large.
Rayleigh's solution
[edit]Two problems that were originally considered by Lord Rayleigh is given here.[5]
Fluids with heat conduction and without viscosity
[edit]The problem when viscosity is neglected but heat conduction is allowed is of significant interest in astrophysical context due to presence of other heat exchange mechanisms such as radiative heat transfer, electron heat transfer in plasmas, etc.[8] Neglect of viscosity means viscous forces in the momentum equation and the viscous dissipation in the energy equation disappear. Hence the first integral of the governing equations are simply given by
All the required ratios can be expreses in terms of immediately,
By eliminating from the last two equations, one can obtain equation , which can be integrated. It turns out there is no continuous solution for strong shock waves, precisely when[8]
for this condition becomes
Fluids with viscosity and without heat conduction
[edit]Here continuous solutions can be found for all shock wave strengths. Further, here the entropy increases monotonically across the shock wave due to the absence of heat conduction. Here the first integrals are given by
One can eliminate the viscous terms in the last two equations and obtain a relation between and . Substituting this back in any one of the equations, we obtain an equation for , which can be integrated.
See also
[edit]References
[edit]- ^ Becker, R. (1922). Stosswelle und detonation. Zeitschrift für Physik, 8(1), 321-362.
- ^ Morduchow, M., & Libby, P. A. (1949). On a complete solution of the one-dimensional flow equations of a viscous, heat-conducting, compressible gas. Journal of the Aeronautical Sciences, 16(11), 674-684.
- ^ Roy, M. , Compt. Rend. Acad. Sei., 218, 813 (1944
- ^ Thomas, L. H. , Jour. Chem. Phys. , 1 2, 449 (1944)
- ^ a b Rayleigh, L. (1910). Aerial plane waves of finite amplitude. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 84(570), 247-284.
- ^ a b Taylor, G. I. (1910). The conditions necessary for discontinuous motion in gases. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 84(571), 371-377.
- ^ Taylor, G. I., & Maccoll, J. W. (1935). The mechanics of compressible fluids. Durand, Aerodynamic theory, 3.
- ^ a b c d e Zel'Dovich, Y. B., & Raizer, Y. P. (2002). Physics of shock waves and high-temperature hydrodynamic phenomena. Courier Corporation.
- ^ Weiss, A. D., Vera, M., Liñán, A., Sánchez, A. L., & Williams, F. A. (2018). A novel formulation for unsteady counterflow flames using a thermal-conductivity-weighted coordinate. Combustion Theory and Modelling, 22(1), 185-201.
- ^ Landau, L. D., & Lifshitz, E. M. (2013). Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6 (Vol. 6). Elsevier.
- ^ Clavin, P., & Searby, G. (2016). Combustion waves and fronts in flows: flames, shocks, detonations, ablation fronts and explosion of stars. Cambridge University Press.