Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.

Filling box flows in porous media

Abstract: We report upon a theoretical and experimental investigation of a porous medium ‘filling box’ flow by specifically examining the details of the laminar descending plume and its outflow in a control volume having an impermeable bottom boundary and sidewalls. The plume outflow initially comprises a pair of oppositely directed gravity currents. The gravity currents propagate horizontally until they reach the lateral sidewalls at . The flow then becomes of filling box type, with a vertically ascending ‘first front’ separating discharged plume fluid below from ambient fluid above. The flow details are described analytically by first deriving a new similarity solution for Darcy plumes with , where is the Peclet number. From the similarity solution so obtained, we then derive expressions for the plume volume flux and mean reduced gravity as functions of the vertical distance from the source. Regarding the plume outflow, a similarity solution adopted from Huppert & Woods (J. Fluid Mech., vol. 292, 1995, pp. 55–69) describes the height and front speed of the gravity currents, whereas a semi-implicit finite difference scheme is used to predict the first front elevation versus time and horizontal distance. As with high-Reynolds-number filling box flows, that studied here is an example of a coupled problem: the gravity current source conditions are prescribed by the plume volume flux and mean reduced gravity. Conversely, discharged plume fluid may be re-entrained into the plume, be it soon or long after reaching the bottom impermeable boundary. To corroborate our model predictions, analogue laboratory experiments are performed with fresh water and salt water as the working fluids. Our experiments consider as independent variables the porous medium bead diameter and the plume source volume flux and reduced gravity. Predictions for the gravity current front position and height compare favourably against analogue measured data. Good agreement is likewise noted when considering either the mean elevation or the profile of the first front. Results from this study may be adopted in modelling geological plumes. For example, our equations can be used to predict the time required for discharged plume fluid to return to the point of injection in the case of aquifers closed on the sides and below by impermeable boundaries.

On natural convection from a vertical plate with a prescribed surface heat flux in porous media

Abstract: This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x2)μ), where μ is a constant andx is the distance along the surface. It is shown that for μ > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when μ ⩽ -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely μ < -1/2 and μ = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of μ.

Similarity solutions for the two-dimensional nonstationary ideal MHD equations

Abstract: A similarity analysis of the non-linear two-dimensional non-stationary ideal MHD equations is presented. In the case of a magnetic field perpendicular to the isentropic motion of the plasma, the authors establish the complete Lie algebra of infinitesimal symmetries. The laws of conservation are mentioned. The similarity method for partial differential equations as a procedure for reducing the number of independent variables is applied repeatedly. Finally, they obtain systems of ordinary differential equations for similarity solutions of the MHD equations considered.

Mixed convection near a non-orthogonal stagnation point flow on a vertical plate with uniform surface heat flux

Abstract: The interaction of a buoyancy induced mixed convection flow and a free stream impinging at some angle of incidence on a vertical flat plate with a prescribed surface heat flux is studied in this paper. The similarity equations are numerically solved for some values of the governing parameters. It is found that the buoyancy force and non-orthogonal mechanisms act to either reinforce or oppose one another. The stagnation point (separation point) is shifted at the left or at the right of the origin and it depends upon the balance between obliqueness and thermal effects.

Linear stability of a gas boundary layer flowing past a thin liquid film over a flat plate

Abstract: The flow of a gas stream past a flat plate under the influence of rainfall is investigated. As raindrops sediment on the flat plate, they coalesce to form a water lm that flows under the action of shear from the surrounding gas stream. In the limit of (a) large Reynolds number, Re, in the gas phase, (b) small rainfall rate, _, compared to the free-stream velocity, U1, and (c) small lm thickness compared to the thickness of the boundary layer that surrounds it, a similarity solution is obtained that predicts growth of the liquid lm like x 3=4 ; x denotes dimensionless distance from the leading edge. The flow in the gas stream closely resembles the Blasius solution, whereas viscous dissipation dominates inside the lm. Local linear stability analysis is performed, assuming nearly parallel base flow in the two streams, and operating in the tripledeck regime. Two distinct families of eigenvalues are identied, one corresponding to the well-known Tollmien{Schlichting (TS) waves that originate in the gas stream, and the other corresponding to an interfacial instability. It is shown that, for the air{water system, the TS waves are convectively unstable whereas the interfacial waves exhibit a pocket of absolute instability, at the streamwise location of the applied disturbance. Moreover, it is found that as the inverse Weber number (We 1 ) increases, indicating the increasing eect of surface tension compared to inertia, the pocket of absolute instability is translated towards larger distances from the leading edge and the growth rate of unstable waves decreases, until a critical value is reached, We 1 We 1 c , beyond which the family of interfacial waves becomes convectively unstable. Increasing the inverse Froude number (Fr 1 ), indicating the increasing eect of gravity compared to inertia, results in the pocket of absolute instability shrinking until a critical value is reached, Fr 1 Fr 1 c , beyond which the family of interfacial waves becomes convectively unstable. As We 1 and Fr 1 are further increased, interfacial waves are eventually stabilized, as expected. In this context, increasing the rainfall rate or the free-stream velocity results in extending the region of absolute instability over most of the airfoil surface. Owing to this behaviour it is conjectured that a global mode that interacts with the boundary layer may arise at the interface and, eventually, lead to three-dimensional waves (rivulets), or, under extreme conditions, even premature separation.