{"id":57,"date":"2022-12-30T18:15:05","date_gmt":"2022-12-30T18:15:05","guid":{"rendered":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/chapter\/rayleigh-number\/"},"modified":"2023-01-03T03:18:42","modified_gmt":"2023-01-03T03:18:42","slug":"rayleigh-number","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/chapter\/rayleigh-number\/","title":{"raw":"5.2 Rayleigh Number","rendered":"5.2 Rayleigh Number"},"content":{"raw":"
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The mixed convection ratio is useful for determining the strength of the density driving force relative to the forced convection driving force. In systems where there is initially no flow, but a concentration or temperature difference exists, the onset of instability is determined by the value of a nondimensional number called the Rayleigh number (Ra<\/em>). It was named in the honor of Lord Rayleigh, who published some of the earliest work in classical fluid mechanics dealing with unstable flows in his paper \u201cOn convection currents in a horizontal layer of fluid when the higher temperature is on the <\/em>underside<\/em>\u201d <\/em>(Rayleigh, 1916). <\/em>The Rayleigh number is the ratio between buoyancy-driven forces that drive free convection to forces (caused by diffusion and dispersion) that act to dissipate it, (e.g., Simmons et al., 2001). For a system that extends infinitely in the horizontal direction, with impermeable top and bottom boundaries between which a density difference of \u03c1<\/em> is maintained (Figure\u00a013) the Rayleigh number is defined as shown in Equation\u00a026.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle Ra=\\frac{\\Delta \\rho gkH}{\\mu D}[\/latex]<\/td>\r\n(26)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u2206\u03c1<\/em><\/td>\r\n=<\/td>\r\ndensity difference across the layer (M\/L3<\/sup>), e.g., kg\u00a0m\u22123<\/sup><\/td>\r\n<\/tr>\r\n
H<\/em><\/td>\r\n=<\/td>\r\nthickness of the layer (L), e.g., m<\/td>\r\n<\/tr>\r\n
D<\/em><\/td>\r\n=<\/td>\r\ndiffusion coefficient (L2<\/sup>\/T), e.g., m2<\/sup> s\u22121<\/sup><\/td>\r\n<\/tr>\r\n
<\/td>\r\n<\/td>\r\nfor solutes, D<\/em> = nD<\/em>C<\/em><\/sub>, porosity times molecular diffusion coefficient<\/td>\r\n<\/tr>\r\n
<\/td>\r\n<\/td>\r\nfor heat, D<\/em> = D<\/em>T<\/em><\/sub>, thermal diffusion coefficient<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

\"Figure<\/p>\r\n

Figure\u00a0<\/strong>13<\/strong> -\u00a0Schematic representation of the horizontally infinite system used to define the Rayleigh number, with impermeable top and bottom boundaries where a density difference \u2206\u03c1 <\/em><\/span>exist over the height H.<\/em> The diagonal line indicates a linear variation in fluid density between the bottom and top boundaries with higher density at the top<\/span>.<\/span><\/p>\r\n

It is important to consider the different meanings of D<\/em> in Equation\u00a026 for solutes and heat as discussed in subsection 4.3<\/a>. For solutes, diffusive transport can take place only through the water-filled pore space and therefore includes the effects of porosity (the fraction of the cross-sectional area across which diffusion occurs due to the presence of solids) and tortuosity (the longer transport pathway compared to diffusion in free water due to the tortuous path lines caused by the irregular geometry of the pore space). For heat, conduction is both through the water-filled pores and the rock itself, and therefore D<\/em>T<\/em><\/sub> is an averaged thermal diffusivity determined by the thermal characteristics of the water and the rock (Ingebritsen et al., 2006).<\/p>\r\n

For Rayleigh numbers greater than some critical Rayleigh number Ra<\/em>c<\/em><\/sub>, gravity induced instability will occur in the form of waves in the boundary layer that develop into fingers or plumes. These fingers sink downward under gravitational influence. This critical Rayleigh number defines the transition between diffusive transport (at lower than critical Rayleigh numbers) and free convective transport by density-driven fingers (at higher than critical Rayleigh numbers). In the most basic sense, for Ra<\/em> less than Ra<\/em>c<\/em><\/sub>,<\/em> the system is stable. For Ra<\/em> greater than Ra<\/em>c<\/em><\/sub>,<\/em> the system is unstable. For infinitely long, parallel domains, with upper and lower boundaries that are impermeable and remain at a constant temperature or concentration), and no forced horizontal advection, the critical Rayleigh number is Ra<\/em>c<\/em><\/sub>\u00a0=\u00a04\u03c0<\/em>2<\/sup>. For other boundary condition types, different values of Ra<\/em>c <\/em><\/sub>apply (Nield, 1968; Nield and Bejan, 2006).<\/p>\r\n

Table\u00a02 summarizes the main differences between groundwater systems and the experimental settings in classical fluid mechanics where convection theory was developed. It is immediately evident that there are significant differences between the physical conditions in these systems. These are also reflected in the nature of the implicit and explicit assumptions that are made, or can be reasonably made, in each case. An obvious problem arises when theoretical work developed under idealized and simplified conditions is applied to more complex settings that are typically encountered in groundwater applications.<\/p>\r\n

Table<\/strong>\u00a0<\/strong>2<\/strong>\u00a0-\u00a0Some of the explicit and implicit assumptions made in the study of free convection systems in both traditional fluid mechanics (left column) and the study of groundwater hydrology (right column).<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
Traditional fluid mechanics <\/strong><\/td>\r\nGroundwater hydraulics<\/strong><\/td>\r\n<\/tr>\r\n
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    \r\n \t
  • Steady-state assumptions<\/li>\r\n \t
  • Homogeneous layers<\/li>\r\n \t
  • Length scale = layer thickness<\/li>\r\n \t
  • Simple chemistry \/ fluids<\/li>\r\n \t
  • Molecular diffusion<\/li>\r\n \t
  • Rayleigh number predicted a priori<\/li>\r\n \t
  • Known Ra<\/em>c<\/em><\/sub><\/li>\r\n \t
  • Laboratory scale, simple boundary conditions<\/li>\r\n<\/ul>\r\n<\/td>\r\n
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