394 research outputs found
The spatial stability of a class of similarity solutions
The spatial stability of a class of exact similarity solutions of the NavierâStokes equations whose longitudinal velocity is of the form xfâ˛(y), where x is the streamwise coordinate and fâ˛(y) is a function of the transverse, crossâstreamwise, coordinate y only, is determined. These similarity solutions correspond to the flow in an infinitely long channel or tube whose surface is either uniformly porous or moves with a velocity linear in x. Small perturbations to the streamwise velocity of the form x^Îťgâ˛(y) are assumed, resulting in an eigenvalue problem for Îť which is solved numerically. For the porous wall problem, it is shown that similarity solutions in which fâ˛(y) is a monotonic function of y are spatially stable, while those that are not monotonic are spatially unstable. For the acceleratingâwall problem, the interpretation of the stability results is not unambiguous and two interpretations are offered. In one interpretation the conclusions are the same as for the porous problemâmonotonic solutions are stable; the second interpretation is more restrictive in that some of the monotonic as well as the nonmonotonic solutions are unstable
Analysis of the Brinkman equation as a model for flow in porous media
The fundamental solution or Green's function for flow in porous media is determined using
Stokesian dynamics, a molecular-dynamics-like simulation method capable of describing the
motions and forces of hydrodynamically interacting particles in Stokes flow. By evaluating the
velocity disturbance caused by a source particle on field particles located throughout a
monodisperse porous medium at a given value of volume fraction of solids ø, and by
considering many such realizations of the (random) porous medium, the fundamental solution
is determined. Comparison of this fundamental solution with the Green's function of the
Brinkman equation shows that the Brinkman equation accurately describes the flow in porous
media for volume fractions below 0.05. For larger volume fractions significant differences
between the two exist, indicating that the Brinkman equation has lost detailed predictive value,
although it still describes qualitatively the behavior in moderately concentrated porous media.
At low ø where the Brinkman equation is known to be valid, the agreement between the
simulation results and the Brinkman equation demonstrates that the Stokesian dynamics
method correctly captures the screening characteristic of porous media. The simulation results
for ø ⼠0.05 may be useful as a basis of comparison for future theoretical work
The sedimentation rate of disordered suspensions
An explicit expression for the sedimentation velocity at low particle Reynolds number in a concentrated suspension is derived and evaluated for two different approximations to the hydrodynamic interactions: a strict pairwise additive approximation and a far-field, or RotneâPrager, approximation. It is shown that the simple RotneâPrager approximation gives a very accurate prediction for the sedimentation velocity of random suspensions from the dilute limit all the way up to close packing. The pairwise additive approximation, however, fails completely, predicting an aphysical negative sedimentation velocity above a volume fraction Ď â 0.23. The explanation for these different behaviors is shown to be linked to the "effective medium" behavior of the suspensions. It is shown analytically and by Stokesian dynamics simulation that a suspension of neutrally buoyant particles may be modeled as a homogeneous fluid with an effective viscosity, but a sedimenting suspension cannot. As a result, the RotneâPrager approximation actually captures the correct features of the many-body interactions in sedimentation. An analytical expression for the sedimentation rate, which is in good agreement with experiment, is obtained using the PercusâYevick hard-sphere distribution function
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Advanced Techniques for Reservoir Simulation and Modeling of Non-Conventional Wells
This project targets the development of (1) advanced reservoir simulation techniques for modeling non-conventional wells; (2) improved techniques for computing well productivity (for use in reservoir engineering calculations) and well index (for use in simulation models), including the effects of wellbore flow; and (3) accurate approaches to account for heterogeneity in the near-well region
Triangle based TVD schemes for hyperbolic conservation laws
A triangle based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second order accuracy. Numerical results for linear advection and Burgers' equation are presented
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