Viscous Case Research Articles

Stability of plane-parallel vibrational flow in a two-layer system

The stability of the interface separating two immiscible incompressible fluids of different densities and viscosities is considered in the case of fluids filling a cavity which performs horizontal harmonic oscillations. There exists a simple basic state which corresponds to the unperturbed interface and plane-parallel unsteady counter flows; the properties of this state are examined. A linear stability problem for the interface is formulated and solved for both (a) inviscid and (b) viscous fluids. A transformation is found which reduces the linear stability problem under the inviscid approximation to the Mathieu equation. The parametric resonant regions of instability associated with the intensification of capillary-gravity waves at the interface are examined and the results are compared to those found in the viscous case in a fully numerical investigation.

Direct simulation Monte Carlo method for molecular and transitional flow regimes in vacuum components with static and moving surfaces

The gas flow through a vacuum component can be continuous, transitional, or molecular depending on the pressure range and geometries involved. Dilute gas flows in molecular and transitional regimes were simulated using the direct simulation Monte Carlo method. The computational model takes as data combination of pressure, temperature of gas, type of gas, and geometry of the vacuum components. The results of the computations are gasflow, local velocities, and molecular density distribution. The model was first tested and validated for several simple geometries, such as circular and rectangular finite length tubes with static and moving walls, for different flow conditions. The Couette flow problem is treated in all the transition regime, from the molecular regime to the beginning of the viscous regime. The model was found to agree well with other published results, and allowed us to better understand flow phenomena for the different regimes from the molecular case to the viscous case. This first study was conducted to develop, in the future, a model that we can adapt to more complicated geometries of vacuum components, such as stages of turbomolecular or molecular pumps, to predict the flow rate from molecular to transition flow regimes.

Wave structure in Stokes' second problem for a dipolar fluid with nonclassical heat conduction

The classical heat conduction law of Fourier associates an infinite speed of propagation to a thermal disturbance in a material body. Such behavior is a violation of the causality principle. In recent years, several modifications of Fourier's heat law have been proposed. In this work, a modified form of Fourier's heat law, based on the Maxwell-Cattaneo-Fox (MCF) model, is used to analyze the heat conduction effects in Stokes' second problem for a dipolar fluid. The structure of the waves and the influence of the dipolar constants on the velocity field is investigated. These results are then compared to the viscous fluid case. In addition, the displacement thickness and skin friction at the plate are determined.

Accretion Disks around Young Objects. I. The Detailed Vertical Structure

We discuss the properties of an accretion disk around a star with parameters typical of classical T Tauri stars (CTTS), and with the average accretion rate for these disks. The disk is assumed steady and geometrically thin. The turbulent viscosity coefficient is expressed using the alpha prescription and the main heating mechanisms considered are viscous dissipation and irradiation by the central star. The energy is transported by radiation, turbulent conduction and convection. We find that irradiation from the central star is the main heating agent of the disk, except in the innermost regions, R less than 2 AU. The irradiation increases the temperature of the outer disk relative to the purely viscous case. As a consequence, the outer disk (R larger than 5 AU) becomes less dense, optically thin and almost vertically isothermal, with a temperature distribution T proportional to R^{-1/2}. The decrease in surface density at the outer disk, decreases the disk mass by a factor of 4 respect to a purely viscous case. In addition, irradiation tends to make the outer disk regions stable against gravitational instabilities.

A model for cytoplasmic rheology consistent with magnetic twisting cytometry

Magnetic twisting cytometry is gaining wide applicability as a tool for the investigation of the rheological properties of cells and the mechanical properties of receptor—cytoskeletal interactions. Current technology involves the application and release of magnetically induced torques on small magnetic particles bound to or inside cells, with measurements of the resulting angular rotation of the particles. The properties of purely elastic or purely viscous materials can be determined by the angular strain and strain rate, respectively. However, the cytoskeleton and its linkage to cell surface receptors display elastic, viscous, and even plastic deformation, and the simultaneous characterization of these properties using only elastic or viscous models is internally inconsistent. Data interpretation is complicated by the fact that in current technology, the applied torques are not constant in time, but decrease as the particles rotate. This paper describes an internally consistent model consisting of a parallel viscoelastic element in series with a parallel viscoelastic element, and one approach to quantitative parameter evaluation. The unified model reproduces all essential features seen in data obtained from a wide variety of cell populations, and contains the pure elastic, viscoelastic, and viscous cases as subsets.

Time Scalings of Entropy Transfer Process in Free Convection

To study one elementary process of entropy cascade in hard turbulence, we propose a shell model which governs the temperature gradient rotated through the angle \(+\frac{\pi}{2}, {\mib \chi} \equiv (\partial_y T, -\partial_x T)\), and the velocity gradient, i.e., strain rates. This model is based on the correspondence of the dynamics of χ in a two-dimensional free convection system to that of the vorticity in a three-dimensional Navier-Stokes system. We obtain the following results: 1) A steady solution which satisfies Bolgiano-Obukhov (BO) scaling in the viscous case exists. 2) After a small disturbance of large scale is added to the steady solution, time series of the modes in the inertial range become similar to each other under the proper scaling, and each mode has its own characteristic time predicted by BO scaling. 3) When the same disturbance is added to the null state, the modes also evolve similarly in the inertial range. The characteristic times of the modes are, however, explained by a dimensio...

Supersonic Transitional Airfoil Shapes of Minimum Drag

Feasibility of a procedure incorporating transition considerations in optimizing total drag has been demonstrated. The Schittkowski algorithm was modie ed to demonstrate the approach on two-dimensional airfoils. Cubic spline basis functions were used to describe the airfoils, and total drag (wave 1 friction) was minimized under the constraint of a e xed airfoil area. Reynolds numbers were assumed such that the airfoil was transitional, generally with the forward portion laminar and the aft turbulent. The transition locus was calculated using the fast transition prediction module, which provides rapid computation of the Tollmien ‐ Schlichting wave amplie cation factor N and estimates transition point by the e N method. The laminar friction drag was evaluated using self-similar solutions of the boundary-layer equations. The friction drag for the turbulent portion was computed assuming a one-seventh velocity proe le. With this framework, the total drag was a function of the functional xtr, the streamwise location of transition. As a validation of the method, the algorithm gave the correct global optimum for the inviscid case, which is the parabolic arc proe le. For the viscous case, signie cantly different locations of the maximum thickness led to only small differences in the minimum total drag. In addition, the drag reductions from the optimal inviscid parabolic proe le were about 10%. Convergence with respect to the number of spline knots was achieved. Important challenges were met to reduce the effect of truncation errors in the numerical approximation of differentiations such as those used in the evaluation of the Hessian matrix. Despite these dife culties, generalization of the approach to treat ine nite yawed and swept wings accounting for suction, crosse ow instabilities, and vortex drag appears feasible.

Preconditioned Multigrid Methods for Compressible Flow Calculations on Stretched Meshes

Efficient preconditioned multigrid methods are developed for both inviscid and viscous flow applications. The work is motivated by the mixed results obtained using the standard approach of scalar preconditioning and full coarsened multigrid, which performs well for Euler calculations on moderately stretched meshes but is far less effective for turbulent Naiver–Stokes calculations, when the cell stretching becomes severe. In the inviscid case, numerical studies of the preconditioned Fourier footprints demonstrate that a block-Jacobi matrix preconditioner substantially improves the damping and propagative efficiency of Runge–Kutta time-stepping schemes for use with full coarsened multigrid, yielding computational savings of approximately a factor of three over the standard approach. In the viscous case, determination of the analytic expressions for the preconditioned Fourier footprints in an asymptotically stretched boundary layer cell reveals that all error modes can be effectively damped using a combination of block-Jacobi preconditioning and a J-coarsened multigrid strategy, in which coarsening is performed only in the direction normal to the wall. The computational savings using this new approach are roughly a factor of 10 for turbulent Navier–Stokes calculations on highly stretched meshes.

Direct simulations of turbulent flow in a pipe rotating about its axis

Flow in a circular pipe rotating about its axis, at low Reynolds number, is investigated. The simulation is performed by a finite difference scheme, second-order accurate in space and in time. A non-uniform grid in the radial direction yields accurate solutions with a reasonable number of grid points. The numerical method has been tested for the non-rotating pipe in the limit ν→0 to prove the energy conservation properties. In the viscous case a grid refinement check has been performed and some conclusions about drag reduction have been reached. The mean and turbulent quantities have been compared with the numerical and experimental non-rotating pipe data of Eggels et al. (1994a, b). When the pipe rotates, a degree of drag reduction is achieved in the numerical simulations just as in the experiments. Through the visualization of the vorticity field the drag reduction has been related to the modification of the vortical structures near the wall. A comparison between the vorticity in the non-rotating and in the high rotation case has shown a spiral motion leading to the transport of streamwise vorticity far from the wall.

Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow

The growth of the gradient of a scalar temperature in a quasigeostrophic flow is studied numerically in detail. We use a flow evolving from a simple initial condition which was regarded by Constantin et al. as a candidate for a singularity formation in a finite time. For the inviscid problem, we propose a completely different interpretation of the growth, that is, the temperature gradient can be fitted equally well by a double-exponential function of time rather than an algebraic blowup. It seems impossible to distinguish whether the flow blows up or not on the basis of the inviscid computations at hand. In the viscous case, a comparison is made between a series of computations with different Reynolds numbers. The critical time at which the temperature gradient attains the first local maximum is found to depend double logarithmically on the Reynolds number, which suggests the global regularity of the inviscid flow.

Multiresolution Schemes for Conservation Laws with Viscosity

This paper presents multiresolution schemes for the efficient numerical solution of one-dimensional conservation laws with viscosity. The method, originally developed by A. Harten (Commun. Pure Appl. Math.,to appear) for hyperbolic conservation laws, computes the cell average multiresolution representation of the solution which provides much information about the solution's regularity. As a consequence, the possibly expensive ENO (essentially nonoscillatory) reconstruction as well as numerous flux computations are performed only near discontinuities, and thereby the numerical solution procedure becomes considerably more efficient. The multiresolution scheme is also expected to “follow” possibly unsteady irregularities from one time step to the next. When viscosity is added, predicting the location of the irregularity becomes a problem of estimating the change in shock thickness. To this end, we derive shock width estimates for our 1D prototype equations, which, when combined with the stability restriction of the numerical scheme, provide a reliable mechanism for enlarging the original multiresolution stencil. The numerical experiments for scalar conservation laws indicate the feasibility of multiresolution schemes for the viscous case as well.

A Stochastic Navier-Stokes Equation for the Vorticity of a Two-Dimensional Fluid

The Navier-Stokes equation for the vorticity of a viscous and incompressible fluid in $\mathbf{R}^2$ is analyzed as a macroscopic equation for an underlying microscopic model of randomly moving vortices. We consider $N$ point vortices whose positions satisfy a stochastic ordinary differential equation on $\mathbf{R}^{2N}$, where the fluctuation forces are state dependent and driven by Brownian sheets. The state dependence is modeled to yield a short correlation length $\varepsilon$ between the fluctuation forces of different vortices. The associated signed point measure-valued empirical process turns out to be a weak solution to a stochastic Navier-Stokes equation (SNSE) whose stochastic term is state dependent and small if $\varepsilon$ is small. Thereby we generalize the well known approach to the Euler equation to the viscous case. The solution is extended to a large class of signed measures conserving the total positive and negative vorticities, and it is shown to be a weak solution of the SNSE. For initial conditions in $L_2(\mathbf{R}^2, dr)$ the solutions are shown to live on the same space with continuous sample paths and an equation for the square of the $L_2$-norm is derived. Finally we obtain the macroscopic NSE as the correlation length $\varepsilon \rightarrow 0$ and $N \rightarrow \infty$ (macroscopic limit), where we assume that the initial conditions are sums of $N$ point measures. As a corollary to the above results we obtain the solution to a bilinear stochastic partial differential equation which can be interpreted as the temperature field in a stochastic flow.

A second order finite element method for the solution of the transonic Euler and Navier‐Stokes equations

AbstractThe numerical solution of the compressible Euler and Navier‐Stokes equations in primitive variables form requires the use of artificial viscosity or upwinding. Methods that are first‐order‐accurate are too dissipative and reduce the effective Reynolds number substantially unless a very fine grid is used. A first‐order finite element method for the solution of the Euler and Navier‐Stokes equations can be constructed by adding Laplacians of the primitive variables to the governing equations. Second‐order schemes may require a fourth‐order dissipation and higher‐order elements. A finite element approach is proposed in which the fourth‐order dissipation is recast as the difference of two Laplacian operators, allowing the use of bilinear elements. The Laplacians of the primitive variables of the first‐order scheme are thus balanced by additional terms obtained from the governing equations themselves, tensor identities or other forms of nodal averaging. To demonstrate formally the accuracy of this scheme, an exact solution is introduced which satisfies the continuity equation identically and the momentum equations through forcing functions. The solutions of several transonic and supersonic inviscid and laminar viscous test cases are also presented and compared to other available numerical data.

Localized disturbances in parallel shear flows

The development of localized disturbances in parallel shear flows is reviewed. The inviscid case is considered, first for a general velocity profile and then in the special case of plane Couette flow so as to bring out the key asymptotic results in an explicit form. In this context, the distinctive differences between the wave-packet associated with the asymptotic behavior of eigenmodes and the non-dispersive (inviscid) continuous spectrum is highlighted. The largest growth is found for three-dimensional disturbances and occurs in the normal vorticity component. It is due to an algebraic instability associated with the lift-up effect. Comparison is also made between the analytical results and some numerical calculations. Next the viscous case is treated, where the complete solution to the initial value problem is presented for bounded flows using eigenfunction expansions. The asymptotic, wave-packet type behaviour is analyzed using the method of steepest descent and kinematic wave theory. For short times, on the other hand, transient growth can be large, particularly for three-dimensional disturbances. This growth is associated with cancelation of non-orthogonal modes and is the viscous equivalent of the algebraic instability. The maximum transient growth possible to obtain from this mechanism is also presented, the so called optimal growth. Lastly the application of the dynamics of three dimensional disturbances in modeling of coherent structures in turbulent flows is discussed.

Falling spheres in polymeric solutions: Limiting results of the two-fluid theory of migration

In this paper we derive a few analytical results for the case of polymeric solutions flowing past a spherical obstacle. The calculations are meant to ascertain the possible importance of flow-induced migration (i.e., of changes in concentration), making use of the recent two-fluid theory of Doi and Milner. In order to obtain solutions in closed form, the problem is suitably linearised; in particular, we have considered both the case of Newtonian behaviour of the polymeric liquid flowing past the sphere, and that of a linearly elastic polymeric gel perturbed by a localised force. The results, though confirming migration effects, indicate that, in these linear limits at least, the concentration change decays quadratically with the reciprocal distance from the centre of the perturbation, i.e., no long “wake” is predicted. Another outcome of these calculations refers specifically to the viscous case. The classical Stokes result is recovered only as a singular solution of the two-fluid problem.

Time domain acoustic absorption: A unified model for linear and nonlinear acoustics

Dissipation of a linear acoustic wave, at frequency ω0, is directly related to the pressure absorption coefficient α(ω). In the viscous case, α(ω)=(4/3η+ηb)ω20/(2ρ0c3). For an initially ω0-sinusoidal large-amplitude nonlinear wave, the absorption coefficient is usually described by a linear part, and by an excess part related to the harmonics from nonlinear distortion. An alternate way to describe the absorption of a nonlinear acoustic wave is to define a total absorption coefficient. This is done in the time domain directly by considering dissipation in the momentum equation, and then proceeding in an energy balance approach. The result gives the instantaneous α for a given point on the wave in space without the need for spectral decomposition. For the viscous case, α(x) = (4/3η + ηb)‖∇2ν‖x0‖/(2ρ0c‖ν0‖), where ν is the particle velocity and x0 is the evaluation site. For harmonic dependence, this reduces to the preceding value α(ω). Similar procedures yield α(x) for heat conduction and relaxation. Examples will be shown for the propagation of linear and nonlinear waves, in water, and in a relaxation-type tissuelike rubber. [Work supported by C. S. Draper Laboratory.]

A study of hydrothermal convection in saturated porous media

Abstract Because of its relevance to many geological and technical problems, hydrothermal convection is investigated here mainly with the aid of numerical models by a systematic analysis of the properties of this type of convection for a range of super-critical Rayleigh numbers. Calculations were performed for two-dimensional models with constant properties in a region of aspect ratio 2. The principal results in the case of temperature fixed at the impermeable top and bottom are the following for the Nusselt number Nu, the cell aspect ratio a, and the boundary layer thickness δ: Nu ≈ 1.7 R0.5, a ≈ 1.3 R−0.4, δc ≈ 0.4 R−0.4 for 2.5 R = R f /R f ∗ and Rf and R f ∗ are the ambient and critical filtration Rayleigh numbers, respectively, relevant to the hydrothermal problem. Beside the usual critical Rayleigh number R f ∗ two additional, new ones have been found: R f ∗∗ which characterizes the appearance of boundary layers, and R f ∗∗∗ which marks the transition to the non-steady state. A number of definitions of the boudary layer are analyzed, and a new definition is proposed based on the condition that the conductive heat flux is equal or greater than the convective one. The initial oscillations after the onset of convection in our model are analyzed and interpreted for the flow in porous media in comparison to the viscous case. Finally a diagram is constructed for the easy evaluation of the numerical accuracy as a function of grid size.

Conduit solitary waves in a visco-elastic medium

Abstract The theory for waves on a buoyant fluid conduit in a more viscous outer fluid is extended to include a visco-elastic outer fluid. The external fluid is treated as a linear Kelvin-type visco-elastic medium and a wave evolution equation is derived. This equation is identical to the purely viscous case with the exception of a new term representing the elastic effects. A conservation law is derived and used in an analytic treatment for a slowly-varying solitary wave (given initially by the exact solution to the purely viscous case) for the case of small, but non-zero, elasticity. The theory shows that the wave amplitude will decay and a shelf, required for the conservation of mass, will develop behind the wave. Numerical solutions of the evolution equation support the analytic approximation. Laboratory experiments show qualitative agreement with the analytic and numerical development. Geophysical applications suggest that these effects may be most important for melt migration in the asthenosphere.

Elastic short wave instability in extrusion flows of viscoelastic liquids

An analysis of the stability to short waves of the flow of concentric coextruded polymeric liquids modeled by upper convected Maxwell models is presented. The flow can be unstable to short waves under various conditions on the elastic parameter. The growth rates for the short wave instability in the elastic case are finite at leading order. The same short wave instability is known to be two orders of magnitude smaller in the purely viscous case with growth rates proportional to α −2 for large wavenumber α. It is argued that this instability could appear in systems with low surface tension, as in polymer-depleted solutions at the walls of a pipe. Some speculative scenarios concerning the appearance of sharkskin after the loss of adhesion, leading to wet slip, are advanced.

Asymmetric ekman dissipation, sloping boundaries and linear baroclinic instability

Abstract The effects of symmetric and asymmetric Ekman dissipation on baroclinic instability, phase speed and wave structure in a linear Eady-like channel model with/without oppositely sloping boundaries are examined. In the absence of sloping boundaries, symmetric Ekman dissipation has a stabilizing tendency for all waves. When the Ekman layers are asymmetric, the viscous asymmetry destabilizes waves by (i) extending the unstable waveband toward both long and short waves, and (ii) increasing their growth rates, compared with viscous symmetric case. When the asymmetric dissipation is very small, the destabilization may result in a frictional instability for short waves which are stable in inviscid case. The viscous asymmetry makes waves dispersive and their structure asymmetric about mid-depth. However, the sense of the viscous asymmetry and the sign of shear do not affect the instability, but do modify the direction of phase propagation and the shape of the wave structure. With asymmetric Ekman dissipati...