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Volume 3, Issue 7, July 1991

Selfdilating viscous fingers in wedgeshaped HeleShaw cells
View Description Hide DescriptionViscous fingering in a wedgeshaped HeleShaw cell is analyzed. The shape of a selfdilating viscous finger is shown to obey a timeindependent nonlinear integrodifferential equation that is solved numerically. The results show that there is a continuum of zerosurfacetension solutions for any wedge angle between 0 and π/2. When surface tension is taken into account only a discrete set of solutions exist. In contrast to the classical case, different branches merge for finite values of the surface tension parameters.

Viscoelastic Poiseuille flow through a curved channel: A new elastic instability
View Description Hide DescriptionThe linear stability of the inertialess, pressuredriven Poiseuille flow of an OldroydB fluid through a slightly curved channel is considered. The flow is shown to be unstable in certain flow parameter regimes. The critical conditions and the structure of the vortex flow at the onset of instability are presented. These results reveal that there is a purely elastic, instability in the flow, and the instability is a stationary mode in contrast to the elastic, oscillatory instability that occurs in TaylorCouette flow [see Larson, Shaqfeh, and Muller, J. Fluid Mech. 218, 573 (1990)]. In addition, the mechanism of the instability is investigated through an examination of the disturbandeenergy equation.

Convergence of Galerkin solutions using Karhunen–Loève expansions of inhomogeneous 1‐D turbulence
View Description Hide DescriptionThe rate of convergence of the Karhunen–Loève expansion of an inhomogeneous, instantaneous random field is compared with that of Fourier expansion in relation to the Reynolds number. The model turbulence is generated by solving the Burgers’ equation with random forcing. The coefficients of the Fourier expansion are determined by a Galerkin solution scheme. The results show obvious superiority of the Karhunen–Loève expansion, especially for high Reynolds number flows.

Observations of pattern evolution in thermal convection with high‐resolution quantitative schlieren imaging
View Description Hide DescriptionHigh‐resolution measurements of the mean temperature gradient field of a fluid undergoing thermal convection in a vertical slot were made. The detection method employs schlieren optics with a linear photodiode array detector. With proper calibration the schlieren signals permit quantitative reconstruction of the varying temperature field along one dimension in the fluid, with a relative temperature resolution of ±0.02 K. The critical applied temperature difference, ΔT _{ b,c }, is determined by extrapolation of the measured convection pattern amplitudes. The measurements permit detailed monitoring of the wave number distribution and its development in time as well as harmonic analysis of the temperature waveform. The stability limits of convective patterns are investigated as a function of forcing and wave number and the observed limits are compared to theoretical predictions. The details of the transitions from conduction to convection and from one convective spatial pattern to another are observed and analyzed as the degree of forcing is varied. The creation and destruction of convective roll cells is observed as is the propagation of convective flow across the slot.

Ideal jets falling under gravity
View Description Hide DescriptionSteady two‐dimensional jets of an inviscid and incompressible fluid emerging from a nozzle and falling under gravity are calculated numerically by series truncation. The nozzle is aimed at an angle β above the horizontal. It is shown that there are flows for all values of β between 0° and 90° and for all values of the Froude number F. Local solutions are constructed to describe the limiting behavior of the flows as F→0 and as F→∞. The problem of a uniform free stream hitting a vertical wall, rising, and forming a jet that falls back upon the oncoming stream is also solved. It is shown that solutions exist only for Froude numbers greater than 2.96.

Stability of an oscillating Kolmogorov flow
View Description Hide DescriptionThe linear stability of unidirectional flow sinusoidal both in a transverse direction and in time is considered. The problem reduces to an infinite algebraic eigenvalue problem. By using continued fractions, it is proved rigorously that the time‐independent flow is unstable to perturbation modes which do not have the periodicity of the basic flow in the transverse direction. Also, instability is proved for the inviscid case, for which the proofs known before do not work even when perturbation modes have the same periodicity as the basic flow. In the case of time‐dependent inviscid flow, exact solutions of a generalized Orr–Sommerfeld equation are found by the separation of continuous variables. Comparison with Galerkin solutions of the eigenvalue problem obtained by the separation of a discrete ‘‘time’’ variable leads to insights into reliability of the Galerkin results for the original, two‐dimensional eigenvalue problem.

Dispersion of suspended particles in turbulent flow
View Description Hide DescriptionThe upward dispersion of heavy particles in suspension in turbulent flow was studied using a numerical model. The interaction between the turbulence and the particle diffusion leads to the formation of a horizontal front (or a ‘‘lutocline’’), across which the diffusion of particles and the propagation of turbulent energy are inhibited. However, as the settling velocity of the particles becomes larger, or as the particle concentration becomes smaller, the interaction weakens, thus suppressing the front formation. One‐dimensional model equations for the problem are solved numerically to calculate the evolution of the particle concentration. A criterion for the formation of the front is proposed and the steady depth of the suspension layer is determined.

Dynamics of fluid mixing in separated flows
View Description Hide DescriptionThe dynamics of the flow in the separated region and near wake of a vertical flat plate was analyzed with the phase‐locked laser‐Doppler technique at Reynolds number Re=2.8×10^{4}. The flow exhibits an intense quasiperiodic component at a Strouhal frequency of St=0.14. The vortex shedding process was resolved into six phases, resulting in a time difference between consecutive phases of Δt=0.02 sec. Interactions between the free shear layers, emanating from the model edges, lead to nonlinear dynamics of the flow. The distributions of the Reynolds stresses indicate that maximal amplitudes in the normal stresses 〈u ’ ^{2}〉 and 〈v ’ ^{2}〉 as well as in the shear stress term 〈u ’ v ’〉 are provoked by positive pressure gradients in the shear layer flows upstream of saddle points. Saddles are quasiperiodically formed by interactions between the two free shear layers. Regions with enhanced Reynolds stresses and saddles are convectively transported within the flow field.

A numerical study of flow separation and reattachment on a blunt plate
View Description Hide DescriptionA two‐dimensional time‐dependent numerical study of separating and reattaching flow over a blunt plate is described. Four Reynolds numbers, Re=150, 250, 300, and 1000, are studied. The first three are in the steady flow regime and calculated values of reattachment lengths compare well with experimental data. For Re=1000, the separated shear layer becomes unsteady with the formation of spanwise vortices. These vortices coalesce and are shed periodically from the reattachment region. Although the resulting flow field is known to be three dimensional, the current two‐dimensional calculation is able to predict important flow properties. Calculated time‐dependent properties such as vortex shedding frequency and convection velocities compare well with experimental data. The present study is a precursor to a three‐dimensional simulation.

A dynamic subgrid‐scale eddy viscosity model
View Description Hide DescriptionOne major drawback of the eddy viscosity subgrid‐scale stress models used in large‐eddy simulations is their inability to represent correctly with a single universal constant different turbulent fields in rotating or sheared flows, near solid walls, or in transitional regimes. In the present work a new eddy viscosity model is presented which alleviates many of these drawbacks. The model coefficient is computed dynamically as the calculation progresses rather than input a p r i o r i. The model is based on an algebraic identity between the subgrid‐scale stresses at two different filtered levels and the resolved turbulent stresses. The subgrid‐scale stresses obtained using the proposed model vanish in laminar flow and at a solid boundary, and have the correct asymptotic behavior in the near‐wall region of a turbulent boundary layer. The results of large‐eddy simulations of transitional and turbulent channel flow that use the proposed model are in good agreement with the direct simulation data.

Subgrid‐scale backscatter in turbulent and transitional flows
View Description Hide DescriptionMost subgrid‐scale (SGS) models for large‐eddy simulations (LES) are absolutely dissipative (that is, they remove energy from the large scales at each point in the physical space). The actual SGS stresses, however, may transfer energy to the large scales (backscatter) at a given location. Recent work on the LES of transitional flows [Piomelli e t a l., Phys. Fluids A 2, 257 (1990)] has shown that failure to account for this phenomenon can cause inaccurate prediction of the growth of the perturbations. Direct numerical simulations of transitional and turbulent channel flow and compressible isotropic turbulence are used to study the backscatter phenomenon. In all flows considered roughly 50% of the grid points were experiencing backscatter when a Fourier cutoff filter was used. The backscatter fraction was less with a Gaussian filter, and intermediate with a box filter in physical space. Moreover, the backscatter and forward scatter contributions to the SGS dissipation were comparable, and each was often much larger than the total SGS dissipation. The SGS dissipation (normalized by total dissipation) increased with filter width almost independently of filter type. The amount of backscatter showed an increasing trend with Reynolds number. In the near‐wall region of the channel, events characterized by strong Reynolds shear stress correlated fairly well with areas of high SGS dissipation (both forward and backward). In compressible isotropic turbulence similar results were obtained, independent of fluctuation Mach number.

Asymptotic behavior of curvature of surface elements in isotropic turbulence
View Description Hide DescriptionThe asymptotic behavior of the curvature of material elements in turbulence is investigated using Lagrangian velocity‐gradient time series obtained from direct numerical simulations of isotropic turbulence. Several material‐element ensembles of different initial curvatures and shapes are studied. It is found that, at long times, the (first five) moments of the logarithm of characteristic curvature (C) and shape factor (S) asymptote to values that are independent of the initial curvature or shape. This evidence strongly suggests that the asymptotic pdf’s of the curvature and shape of material elements are stationary and independent of initial conditions. Irrespective of initial curvature or shape, the asymptotic shape of a material surface is cylindrical with a high probability.

Inverse cascades in three‐dimensional anisotropic flows lacking parity invariance
View Description Hide DescriptionA three‐dimensional flow stirred by an anisotropic body force lacking parity invariance, may be unstable to large‐scale perturbations. This is the anisotropic kinetic alpha (AKA) effect. When an extended range of scales is linearly unstable, an inverse cascade develops. Eventually, the flow is dominated by modes corresponding to the largest available scales.

Analysis of vibrational‐translational energy transfer using the direct simulation Monte Carlo method
View Description Hide DescriptionA new model is proposed for energy transfer between the vibrational and translational modes for use in the direct simulation Monte Carlo method (DSMC). The model modifies the Landau–Teller theory for a harmonic oscillator and the rate of transition is related to an experimental correlation for the vibrational relaxation time. Assessment of the model is made with respect to three different computations: relaxation in a heat bath, a one‐dimensional shock wave, and hypersonic flow over a two‐dimensional wedge. These studies verify that the model achieves detailed balance, and excellent agreement with experimental data is obtained in the shock wave calculation. The wedge flow computation reveals that the usual phenomenological method for simulating vibrational nonequilibrium in the DSMC technique predicts much higher vibrational temperatures in the wake region. Additionally, the numerical performance of the new model is equal to that of the phenomenological scheme.

Shock wave effects on a turbulent flow
View Description Hide DescriptionA parametric study is done to investigate the change in a turbulent flow field caused by the passage of a shock wave. Two parameters are studied: the initial turbulent kinetic energy and the shock wave strength or density jump. A random or turbulent flow field is initiated within a two‐dimensional box. Euler’s equations are then solved using a second‐order accurate Godunov shock capturing method to calculate the change in turbulent structure and flow field parameters caused by the passage of a shock wave through the turbulent field. Two fields were analyzed, a random density field and a random velocity field. The passage of a shock through the random density field caused density and pressure variations that compare very well with experiments. Results of the shock passage through the random velocity field show that the shock causes an amplification in the turbulent kinetic energy of about 2 on a per unit mass basis. Furthermore, the length scale of the turbulent field behind the shock is smaller than that in front of the shock. Energy weighted wave numbers increase by as much as 30%. This change in length scales seems to be in disagreement with some experiments which seem to show larger time scales and larger length scales behind a shock, but in agreement with another experiment. For both results, fields containing strong shocks or weak turbulent fields yield the largest change in flow parameters. The shock wave is also affected by the turbulent field. Increasing the initial turbulent kinetic energy caused a straight shock wave to evolve into a shock containing curves and wrinkles of a size similar to the length scale of the unshocked turbulent field. These curves and wrinkles can lead to the generation of additional flow field oscillations.

The point explosion with heat conduction
View Description Hide DescriptionThe influence of nonlinear heat conduction is investigated for strong point explosions in an ambient gas. An ideal gas equation of state and a heat conductivity depending on temperature and density in power‐law form are assumed. It is shown that two spherical waves are obtained—a shock wave and a heat wave. They have sharp fronts which run at different speeds, in general, and in a relative order depending on parameters and time. Starting from the underlying Lie group symmetry, self‐similar solutions of the problem are discussed in detail; they exist under the assumption that the ambient gas density decays with a given power of the radius. The non‐self‐similar situation, occurring for uniform density, is also considered. In this case, the shock front first runs behind the heat front, but then overtakes it at a certain time t _{1}. For t≫t _{1}, the well‐known hydrodynamic solution of the problem without heat conduction becomes valid, except for a central, almost isobaric region where heat conduction modifies the classical result and keeps the temperature finite. It is shown that this central zone still has the form of a heat wave with a sharp front and evolves self‐similarly, though with a smaller similarity exponent than the global hydrodynamic wave. Analytic results for the central temperature and the radial extension of the conduction dominated zone are given.

The lubrication analysis for two spheres in a two‐dimensional pure‐straining motion
View Description Hide DescriptionThe stresslets of two spheres in a low Reynolds number flow are calculated asymptotically for the case in which the imposed flow field tends, far from the spheres, to a two‐dimensional pure‐straining motion perpendicular to the line of centers. The spheres are assumed to be nearly touching. The analysis is based on a complementary problem in which lubrication theory is used to calculate the flow between two deforming spheres. As well as providing asymptotic estimates of the singular nature of the stresslets, the solution shows an unexpected feature for a lubrication flow, this being the fact that, for one special case, the pressure in the flow field is zero.

A note on memory‐integral contributions to the force on an accelerating spherical drop at low Reynolds number
View Description Hide DescriptionThe hydrodynamic force on a spherical drop that undergoes a translational acceleration in an unbounded fluid at low Reynolds number is considered. The force involves a memory‐integral contribution that is not of the familiar form for a solid sphere. This result, in conjunction with the prior results of Lawrence and Weinbaum [J. Fluid Mech. 171, 208 (1986)] for a nonspherical particle, suggest that the form of the force law for a solid sphere is a very special case that is invalidated if there are any departures in either rigidity or shape from a solid sphere. In this Brief Communication the force on a spherical drop is evaluated for a number of limiting cases, after transforming the result from the Fourier‐transform domain in which it is derived to the time domain.

Stability criteria for flow along a convex wall
View Description Hide DescriptionFlow along a curved wall is susceptible to centrifugal instability. The criterion of Synge [Proc. R. Soc. London Ser. A 1 6 7, 250 (1938)] for the stability of flow between rotating cylinders is generalized to the case of an arbitrary flow along a convex wall. Sufficient conditions for the linear stability of the flow are given, based on the behavior of a function similar to the Rayleigh discriminant.

Axial forcing of an inviscid finite length fluid cylinder
View Description Hide DescriptionCurrent interest in microgravity materials processing has focused attention upon the finite fluid column. This configuration is used in the modeling of float zones. In this Brief Communication, the incompressible inviscid finite length fluid column is subjected to an axial, time‐dependent disturbance. The response of the fluid system and the resulting interface location are determined via Laplace transform methods.