{"id":60,"date":"2022-12-30T18:15:14","date_gmt":"2022-12-30T18:15:14","guid":{"rendered":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/chapter\/wooding-number\/"},"modified":"2023-01-05T05:07:21","modified_gmt":"2023-01-05T05:07:21","slug":"wooding-number","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/chapter\/wooding-number\/","title":{"raw":"5.3 Wooding Number","rendered":"5.3 Wooding Number"},"content":{"raw":"
\r\n

As mentioned in the previous section, application of the Rayleigh number is limited to systems without a background flow field. Wooding (1960) analyzed the case of a boundary layer that grows by diffusion in the presence of a flow that opposes diffusion. Situations like these are found below the bottom of salt lakes that receive groundwater input by upward seepage (Simmons et al., 2002; Simmons et al., 1999; Wooding et al., 1997), salt pans (Bauer et al., 2006) or where groundwater discharges into the ocean (Greskowiak, 2014). A stable boundary layer occurs when the diffusive flux equals the opposing advective flux (Figure\u00a015).<\/p>\r\n

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

Figure\u00a0<\/strong>15<\/strong>\u00a0-\u00a0Schematic representation of a system with a flow component (upward in this image) in the opposite direction as diffusion (downward in this image), where a density difference \u2206\u03c1<\/em> exists over the characteristic boundary layer thickness \u03b4<\/em>. The hatched area is a zone of constant concentration that is higher than the initial concentration in the porous medium. The curved line represents the fluid density distribution below the hatched zone.<\/p>\r\n

Wooding et al. (1997) defined the characteristic thickness \u03b4 <\/em>of the boundary layer as shown in Equation\u00a028.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\delta =\\frac{D}{q_{z}}[\/latex]<\/td>\r\n(28)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

When H<\/em> is replaced by \u03b4<\/em> in Equation\u00a026<\/a>, the boundary layer Rayleigh number, or Wooding number, becomes Equation\u00a029.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle Ra^{\\delta }=\\frac{\\Delta \\rho gk\\delta }{\\mu D}=\\frac{\\Delta \\rho gk}{\\mu q_{z}}[\/latex]<\/td>\r\n(29)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Based on linear perturbation analysis, Wooding (1960) determined the value of the critical boundary Rayleigh number to be [latex]Ra_{cr}^{\\delta }\\approx 7[\/latex]. For values less than 7 which are situations with high q<\/em>z<\/em><\/sub> or low \u0394\u03c1<\/em>, the boundary layer initially increases and then reaches a stable thickness. For values greater than 7 which are situations with low q<\/em>z<\/em><\/sub> or high \u0394\u03c1<\/em>, the boundary layer continues to increase and never stabilizes.<\/p>\r\n

High values of q<\/em>z<\/em><\/sub> tend to stabilize the system, as shown in Figure\u00a016. The colored lines mark the boundary between the stable and unstable regime for a range of values of q<\/em>z<\/em><\/sub> that may be encountered in groundwater systems. Fluxes of q<\/em>z<\/em><\/sub>\u00a0>\u00a00.1\u00a0m\u00a0d\u22121<\/sup> are required to prevent instabilities from amplifying in the most permeable aquifers.<\/p>\r\n

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

Figure\u00a0<\/strong>16<\/strong>\u00a0-\u00a0Boundaries between stable and unstable regimes for different values of q<\/em>z<\/em><\/sub> (in m\u00a0d\u22121<\/sup>), indicated by colored lines. The lines are calculated according to Equation 29 with [latex]Ra=Ra_{cr}^{\\delta }=7[\/latex]. Stable conditions exist to the left, and unstable to the right, of each line marking the boundary between the regimes. Values of q<\/em>z<\/em><\/sub> well above 0.1 m\u00a0d-<\/sup>1<\/sup> would be required to maintain stable conditions in gravels for even small density differences.<\/p>\r\n\r\n<\/div>","rendered":"

\n

As mentioned in the previous section, application of the Rayleigh number is limited to systems without a background flow field. Wooding (1960) analyzed the case of a boundary layer that grows by diffusion in the presence of a flow that opposes diffusion. Situations like these are found below the bottom of salt lakes that receive groundwater input by upward seepage (Simmons et al., 2002; Simmons et al., 1999; Wooding et al., 1997), salt pans (Bauer et al., 2006) or where groundwater discharges into the ocean (Greskowiak, 2014). A stable boundary layer occurs when the diffusive flux equals the opposing advective flux (Figure\u00a015).<\/p>\n

\"Schematic<\/p>\n

Figure\u00a0<\/strong>15<\/strong>\u00a0–\u00a0Schematic representation of a system with a flow component (upward in this image) in the opposite direction as diffusion (downward in this image), where a density difference \u2206\u03c1<\/em> exists over the characteristic boundary layer thickness \u03b4<\/em>. The hatched area is a zone of constant concentration that is higher than the initial concentration in the porous medium. The curved line represents the fluid density distribution below the hatched zone.<\/p>\n

Wooding et al. (1997) defined the characteristic thickness \u03b4 <\/em>of the boundary layer as shown in Equation\u00a028.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle \delta =\frac{D}{q_{z}}\"<\/td>\n(28)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

When H<\/em> is replaced by \u03b4<\/em> in Equation\u00a026<\/a>, the boundary layer Rayleigh number, or Wooding number, becomes Equation\u00a029.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle Ra^{\delta }=\frac{\Delta \rho gk\delta }{\mu D}=\frac{\Delta \rho gk}{\mu q_{z}}\"<\/td>\n(29)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Based on linear perturbation analysis, Wooding (1960) determined the value of the critical boundary Rayleigh number to be \"Ra_{cr}^{\delta }\approx 7\". For values less than 7 which are situations with high q<\/em>z<\/em><\/sub> or low \u0394\u03c1<\/em>, the boundary layer initially increases and then reaches a stable thickness. For values greater than 7 which are situations with low q<\/em>z<\/em><\/sub> or high \u0394\u03c1<\/em>, the boundary layer continues to increase and never stabilizes.<\/p>\n

High values of q<\/em>z<\/em><\/sub> tend to stabilize the system, as shown in Figure\u00a016. The colored lines mark the boundary between the stable and unstable regime for a range of values of q<\/em>z<\/em><\/sub> that may be encountered in groundwater systems. Fluxes of q<\/em>z<\/em><\/sub>\u00a0>\u00a00.1\u00a0m\u00a0d\u22121<\/sup> are required to prevent instabilities from amplifying in the most permeable aquifers.<\/p>\n

\"Graph<\/p>\n

Figure\u00a0<\/strong>16<\/strong>\u00a0–\u00a0Boundaries between stable and unstable regimes for different values of q<\/em>z<\/em><\/sub> (in m\u00a0d\u22121<\/sup>), indicated by colored lines. The lines are calculated according to Equation 29 with \"Ra=Ra_{cr}^{\delta }=7\". Stable conditions exist to the left, and unstable to the right, of each line marking the boundary between the regimes. Values of q<\/em>z<\/em><\/sub> well above 0.1 m\u00a0d–<\/sup>1<\/sup> would be required to maintain stable conditions in gravels for even small density differences.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":8,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-60","chapter","type-chapter","status-publish","hentry"],"part":131,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/60","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":8,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/60\/revisions"}],"predecessor-version":[{"id":290,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/60\/revisions\/290"}],"part":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/parts\/131"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/60\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/media?parent=60"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/chapter-type?post=60"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/contributor?post=60"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/license?post=60"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}