5.2 Rayleigh Number

Vincent E.A. Post; Craig T. Simmons

5.2 Rayleigh Number

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). 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 “On convection currents in a horizontal layer of fluid when the higher temperature is on the underside” (Rayleigh, 1916). 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 ρ is maintained (Figure 13) the Rayleigh number is defined as shown in Equation 26.

$\displaystyle Ra=\frac{\Delta \rho gkH}{\mu D}$

(26)

where:

∆ρ	=	density difference across the layer (M/L³), e.g., kg m⁻³
H	=	thickness of the layer (L), e.g., m
D	=	diffusion coefficient (L²/T), e.g., m² s⁻¹
		for solutes, D = nD_C, porosity times molecular diffusion coefficient
		for heat, D = D_T, thermal diffusion coefficient

Figure used to define the Rayleigh number

Figure 13 – Schematic representation of the horizontally infinite system used to define the Rayleigh number, with impermeable top and bottom boundaries where a density difference ∆ρ exist over the height H. The diagonal line indicates a linear variation in fluid density between the bottom and top boundaries with higher density at the top.

It is important to consider the different meanings of D in Equation 26 for solutes and heat as discussed in subsection 4.3. 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_T is an averaged thermal diffusivity determined by the thermal characteristics of the water and the rock (Ingebritsen et al., 2006).

For Rayleigh numbers greater than some critical Rayleigh number Ra_c, 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 less than Ra_c, the system is stable. For Ra greater than Ra_c, 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_c = 4π². For other boundary condition types, different values of Ra_capply (Nield, 1968; Nield and Bejan, 2006).

Table 2 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.

Table 2 – Some 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).

Traditional fluid mechanics	Groundwater hydraulics
Steady-state assumptions Homogeneous layers Length scale = layer thickness Simple chemistry / fluids Molecular diffusion Rayleigh number predicted a priori Known Ra_c Laboratory scale, simple boundary conditions	Transient Heterogeneous Length scales ambiguous Complex geochemistry / fluid-matrix Dispersion greater than diffusion Cannot determine Rayleigh number a priori Unknown Ra_c Field scale, complex boundary conditions

Simmons and others (2001) and Simmons (2005) described some of the problems using the Rayleigh number in natural groundwater systems. The criteria for extremely simple boundary and layer conditions in traditional fluid mechanics (e.g., an infinite horizontal homogeneous layer with perfect constant concentration upper and lower boundaries) are unlikely to be applicable to most groundwater situations. As Table 2 highlights, one reason for this is that natural systems are not at steady state, the hydraulic properties are heterogeneous, a representative length scale is difficult to define (Rees et al., 2008; Riaz et al., 2006) and the critical Rayleigh number Ra_c is rarely known in a natural groundwater setting.

Despite these limitations, some insight into the role of free convection in groundwater systems can be obtained by observing the range of Rayleigh numbers as a function of ∆ρ and k (Figure 14). Since the range of permeability values for geologic materials is many orders of magnitude, so is the range of possible Rayleigh numbers, which is why the log of Ra was contoured. Figure 14 shows separate plots for solutes and heat because solutes and heat have vastly different values of D (which were taken as D = D_C = 5 × 10⁻¹⁰ and D = D_T = 1.4 × 10⁻⁶ m² s⁻¹, respectively). Moreover, the density range associated with solute concentration differences tends to be larger than for temperature differences, which is why the range on the vertical axes of Figure 14a and Figure 14b differ. Finally, solute gradients can be much higher than common temperature gradients, so different values of H were chosen to draw the graphs of Figure 14a and 14b. Several general conclusions can be drawn from the graphs:

density differences created by temperature differences require a higher permeability for instability to set in than density-differences created by solute concentrations;
for permeable aquifer materials (e.g., sand and gravel), even small density contrast can lead to unstable convection;
the actual value of the density contrast ∆ρ required for onset of unstable conditions is sensitive only within a relatively narrow range of permeabilities (as inferred from the steep vertical trajectory of the white line); and,
for materials with very low permeability (like clay), the density differences caused by variations in solute concentration or temperature in groundwater systems are not large enough to cause instability.

Graph showing range of Rayleigh number

Figure 14 – The Rayleigh number (note the decadal log scale) as a function of permeability k and density contrast ∆ρ for: a) solutes, and b) heat. The white line indicates the combination of k and ∆ρ values where Ra = 4². For dissolved solids, a density difference of 100 kg m⁻³ is roughly equivalent to the difference between fresh water and water with 150 g NaCl kg⁻¹. For heat, at standard pressure, a density difference of 40 kg m⁻³ is roughly equivalent to the difference between fresh water at 4 °C and 98 °C. The labels below the graphs indicate where the k values of common unconsolidated aquifer materials occur on the permeability log scale based on Freeze and Cherry (1979; page 29).

An important caveat must be made for the discussion in the previous paragraph because the use of the solute molecular diffusion coefficient D_C = 5 × 10⁻¹⁰ m² s⁻¹ is not appropriate for aquifers in which solute spreading also occurs by mechanical dispersion (as indicated in Table 2). In solute transport models, the effect of hydrodynamic dispersion is normally described using a dispersion coefficient (D_h), which has the form D_h = αv + D_C, where α is the dispersivity (m) and v the groundwater seepage velocity (m s⁻¹). Because of the directional dependence of the dispersion process, D_h is a tensor (as discussed further in Section 6.1) but this simple expression is useful to gain some insight into the relative magnitude of the molecular diffusion coefficient relative to the hydrodynamic dispersion coefficient. Considering a range of α values from 10⁻² to 100 m, combined with a range of groundwater flow velocities v between 1 and 10 m year⁻¹ yields a corresponding range of 3.17 × 10⁻¹⁰ < D_h < 3.17 × 10⁻⁵ m² s⁻¹.

These values show that solute spreading by mechanical dispersion is more substantial than molecular diffusion and is usually the dominant spreading mechanism in natural groundwater systems. Unfortunately, incorporation of the mechanical dispersion into a Rayleigh number is non-trivial. An alternative Rayleigh number formulation was provided by Hidalgo and Carrera (2009, their Equation 2.18), who took the longitudinal dispersivity (α_L, i.e., the dispersivity in the direction of flow), which reflects the character of the porous medium even though it only influences dispersion if groundwater velocity is greater than zero, and used it as the representative length scale (instead of H) to estimate the onset of instability in an initially stagnant system as shown in Equation 27.

$\displaystyle Ra'=\frac{\Delta \rho gk\alpha _{L}}{\mu (D+\alpha _{L}v)}$

(27)

The onset time of free convection decreased with Ra’ and it was found that the onset time could be up to two orders of magnitude shorter in dispersive systems than in purely diffusive systems. Laboratory experiments have confirmed the importance of mechanical dispersion in controlling the downward solute flux and the finger dimensions (Liang et al., 2018). In these experiments, which were conducted in homogeneous porous media, the dispersivity was a function of the grain size. In the field though, the dispersivity is controlled by the heterogeneity of the geological material, which spans a range of spatial scales. The relevance of mechanical dispersion in field settings, therefore, remains an unresolved issue because convective fingering is influenced by geological heterogeneity in a complex way. This is discussed in Section 6.3.

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