Simple Viscous Flows: from Boundary Layers to the Renormalization Group

The seemingly simple problem of determining the drag on a body moving through a very viscous fluid has, for over 150 years, been a source of theoretical confusion, mathematical paradoxes, and experimental artifacts, primarily arising from the complex boundary layer structure of the flow near the body and at infinity. We review the extensive experimental and theoretical literature on this problem, with special emphasis on the logical relationship between different approaches. The survey begins with the developments of matched asymptotic expansions, and concludes with a discussion of perturbative renormalization group techniques, adapted from quantum field theory to differential equations. The renormalization group calculations lead to a new prediction for the drag coefficient, one which can both reproduce and surpass the results of matched asymptotics

The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number

The hydrodynamic force acting on a rigid spherical particle translating with arbitrary time-dependent motion in a time-dependent flowing fluid is calculated to O(Re) for small but finite values of the Reynolds number, Re, based on the particle's slip velocity relative to the uniform flow. The corresponding expression for an arbitrarily shaped rigid particle is evaluated for the case when the timescale of variation of the particle's slip velocity is much greater than the diffusive scale, a^2/v, where a is the characteristic particle dimension and v is the kinematic viscosity of the fluid. It is found that the expression for the hydrodynamic force is not simply an additive combination of the results from unsteady Stokes flow and steady Oseen flow and that the temporal decay to steady state for small but finite Re is always faster than the t^-½ behaviour of unsteady Stokes flow. For example, when the particle accelerates from rest the temporal approach to steady state scales as t^-2

Computation of Steady Incompressible Flows in Unbounded Domains

In this study we revisit the problem of computing steady Navier-Stokes flows in two-dimensional unbounded domains. Precise quantitative characterization of such flows in the high-Reynolds number limit remains an open problem of theoretical fluid dynamics. Following a review of key mathematical properties of such solutions related to the slow decay of the velocity field at large distances from the obstacle, we develop and carefully validate a spectrally-accurate computational approach which ensures the correct behavior of the solution at infinity. In the proposed method the numerical solution is defined on the entire unbounded domain without the need to truncate this domain to a finite box with some artificial boundary conditions prescribed at its boundaries. Since our approach relies on the streamfunction-vorticity formulation, the main complication is the presence of a discontinuity in the streamfunction field at infinity which is related to the slow decay of this field. We demonstrate how this difficulty can be overcome by reformulating the problem using a suitable background "skeleton" field expressed in terms of the corresponding Oseen flow combined with spectral filtering. The method is thoroughly validated for Reynolds numbers spanning two orders of magnitude with the results comparing favourably against known theoretical predictions and the data available in the literature.Comment: 39 pages, 12 figures, accepted for publication in "Computers and Fluids

The numerical solution of the Navier-Stokes equations for laminar incompressible flow past a paraboloid of revolution

A numerical method is presented for the solution of the Navier-Stokes equations for flow past a paraboloid of revolution. The flow field has been computed for a large range of Reynolds numbers. Results are presented for the skinfriction and the pressure together with their respective drag coefficients. The total drag has been checked by means of an application of the momentum theorem.

The force on a sphere in a uniform flow with small-amplitude oscillations at finite Reynolds number

The unsteady force acting on a sphere that is held fixed in a steady uniform flow with small-amplitude oscillations is evaluated to O(Re) for small Reynolds number Re. Good agreement is shown with the numerical results of Mei, Lawrence & Adrian (1991) up to Re [approximate] 0.5. The analytical result is transformed by Fourier inversion to allow for an arbitrary time-dependent motion which is small relative to the steady uniform flow. This yields a history-dependent force which has an integration kernel that decays exponentially for large time

The force on a bubble, drop, or particle in arbitrary time-dependent motion at small Reynolds number

The hydrodynamic force on a body that undergoes translational acceleration in an unbounded fluid at low Reynolds number is considered. The results extend the prior analysis of Lovalenti and Brady [to appear in J. Fluid Mech. (1993)] for rigid particles to drops and bubbles. Similar behavior is shown in that, with the inclusion of convective inertia, the long-time temporal decay of the force (or the approach to steady state) at finite Reynolds number is faster than the t-1/2 predicted by the unsteady Stokes equations

Tumbling Motion of Elliptical Particles in Viscous Two-Dimensional Flow

The settling dynamics of spherical and elliptical particles in a viscous Newtonian fluid are investigated numerically using a finite difference technique. The terminal velocity for spherical particles is calculated for different system sizes and the extrapolated value for an infinite system size is compared to the Oseen approximation. Special attention is given to the settling and tumbling motion of elliptical particles where their terminal velocity is compared with the one of the surface equivalent spherical particle.Comment: 13 pages, 8 figures (within text), uses IJMPC macros (included

Solutions of the Navier–Stokes Equation at Large Reynolds Number

The problem of two-dimensional incompressible laminar flow past a bluff body at large Reynolds number (R) is discussed. The governing equations are the Navier-Stokes equations. For R = ∞, the Euler equations are obtained. A solution for R large should be obtained by a perturbation of an Euler solution. However, for given boundary conditions, the Euler solution is not unique. The solution to be perturbed is the relevant Euler solution, namely the one which is the Euler limit of the Navier-Stokes solution with the same boundary conditions. For certain semi-infinite or streamlined bodies, the relevant Euler solution represents potential flow. For flow inside a closed domain a theorem of Prandtl states the relevant Euler solution has constant vorticity in each vortex. In many cases it can be determined by simultaneously considering the boundary layer equations. For flow past a bluff body, the relevant Euler solution is not known, although the free streamline flow for which the free streamline detaches smoothly from the body is a likely candidate. Even if this is correct, many unsolved problems remain. Various scalings have to be used for various regions of the flow. Possibilities of scaling for the various regions are discussed here. Special attention is paid to the region near the point of separation. A famous paper by Goldstein asserts that for an adverse smooth pressure gradient, the solution of the boundary layer equations can, in general, not be continued beyond the point of separation. Subsequent attempts by many authors to overcome the difficulty of continuation have failed. A very promising theory, going beyond conventional boundary layer theory, has recently been put forward independently by Sychev and Messiter. They assume that separation takes place in a sublayer whose thickness and length tend to zero as R tends to infinity. The pressure gradient in the sublayer is self-induced and is positive upstream of the point of separation and zero downstream. Their theory does not contradict experiments and numerical calculations, which may be reliable up to, say, R = 100, but it also shows that in this context, 100 may not be regarded as a large Reynolds number. The sublayer has the same scaling in orders of R as the sublayer at the trailing edge of a plate, found earlier by Stewartson and Messiter in studying the matching of the boundary layer solution on the plate with the Goldstein wake solution downstream of the trailing edge