Similarity solutions for two-dimensional steady laminar gravity currents

Abstract

Similarity solutions have been found for steady two-dimensional laminar flow in which dense fluid is emitted upwards from a horizontal plane into a laminar shear flow or into a uniform flow. The solutions also apply to a light fluid released at an upper horizontal surface. The Navier-Stokes equations and the diffusion-advection equation are simplified by making the Boussinesq approximation and the boundary-layer approximation, which here also implies that pressure is hydrostatic.

For an oncoming linear shear flow representing flow near a solid surface, a similarity solution is obtained with depth proportional to $x^{\frac{1}{3}}$ where x is the horizontal coordinate. Horizontal velocity and concentration of dense fluid both increase as $x^{\frac{1}{3}}$, so that the solution represents fluid propagating upstream along the surface, and diffusing vertically to be swept downstream again. Numerical solutions for vertical profiles of velocity and concentration are presented for a Schmidt or Prandtl number σ between 0.71 and infinity. Two alternative sets of boundary conditions are possible. In one set, the pressure above the boundary layer is unchanged but the velocity profile is displaced upwards. In the second, this displacement is forced to be zero with the result that a pressure gradient is generated in the outer flow. These two boundary conditions are known to apply to disturbances in a laminar boundary-layer on horizontal lengthscales respectively greater or smaller than the triple-deck scale.

With a uniform velocity upstream and a stress-free boundary, representing flow at a free surface, similarity solutions exist only for a plume growing downstream from the source of a buoyancy flux B, with depth increasing as x^½ and concentration decreasing as x^−½. When gravity has negligible effects, so that B = 0, the solution is a Gaussian plume. With finite B, there is an adverse gradient of hydrostatic pressure and the plume is decelerated so that it is deeper than in neutral flow. Numerical solutions for σ = 0.71 reveal that there is a maximum buoyancy flux B_crit above which no similarity solution exists. This occurs with a non-zero value of the surface velocity. For B < B_crit it is found that there are in fact two possible solutions. One has surface velocity greater than at the critical flux and tends to the passive Gaussian plume as B → 0. In the other, surface velocity decreases from B_crit, reaching zero at a non-zero value of B. Similar behaviour is found in an asymptotic solution for very large σ.