Second-order Flow Research Articles

Thermally developing microtube gas flow with axial conduction and viscous dissipation

A comparative study of the first and second order slip flow models on the thermal development of dilute gas flow in a microtube with axial conduction and viscous dissipation has been performed. The velocity profiles for the two models have been derived analytically assuming the flow to be fully developed hydrodynamically. The energy equation is solved numerically for a constant wall temperature with velocity slip and temperature jump conditions at the wall. Analytical solutions for the asymptotic values of the mean temperature and Nusselt number are given for the second-order slip flow model in terms of Brinkman number Br, Knudsen number Kn, temperature jump parameter κ, and slip parameter ρ s. The effect of these parameters and the Peclet number Pe on the development of local mean fluid temperature and the local Nusselt number are displayed graphically and compared with the results available in the literature. For the fully developed hydrodynamic condition, the second-order slip model predicts higher velocities in the central region of the flow but the trend is reversed near the wall region. For fixed values of Pe, Kn, κ, and three different Br, the second order model predicts higher fluid temperatures compared with those predicted by the first-order model. For thermally developing flow, the second order model in general predicts significantly higher mean fluid temperatures (maximum of 21% in one case), compared with those predicted by the first-order model. However, the difference in the local Nusselt number between the two slip models is small.

A Large-Eddy Simulation Model for Boundary-Layer Flow Over Surfaces with Horizontally Resolved but Vertically Unresolved Roughness Elements

Accurate prescription of ground-level boundary conditions is a key requirement for large-eddy simulation (LES) of atmospheric boundary-layer (ABL) flow. When the lower boundary is a homogeneously rough surface, the Monin–Obokhov (MO) similarity theory is a well-tested approach for specification of the boundary fluxes. This approach requires an empirically calibrated hydrodynamic roughness length to represent the aggregate effects of all flow-resisting surface details at unresolved scales, and a displacement height to represent a vertical shift in the velocity profiles. For cases in which the surface height distribution varies slowly enough in the horizontal direction that it is spatially resolvable in the LES, but where the height itself is small enough to fall below the first vertical grid point, special challenges arise. First, it is shown herein that prescription of a horizontally varying displacement height, in the context of MO similarity, is insufficient to produce realistic results in this unique application. A subsequent approach, which is relatively easy to implement, and can be used in LES without the need to use a “terrain-following” coordinate system or computationally expensive alternatives such as local grid refinement, is proposed. The approach is based on a form drag expressed in terms of the frontal area and the corresponding incoming momentum flux of the flow. The frontal area is evaluated in terms of the flow-normal gradient of the resolved height distribution. The proposed model, named the surface gradient-based drag (SGD) model, is applied to a variety of rough surfaces for which appropriate data exist. These surfaces are horizontally resolved, but vertically unresolved. The resulting mean velocity distributions are compared with available data and good agreement is observed. In addition, we apply the SGD model in flow over two synthetic multiscale surfaces with prescribed, power-law surface-height spectra. For these cases, first- and second-order flow statistics agree with established results. Future perspectives are outlined.

Optimal Motorway Traffic Flow Control Involving Variable Speed Limits and Ramp Metering

The impact of variable speed limits (VSL) on aggregate traffic flow behaviour on motorways is shown to bear similarities to the impact of ramp metering, in particular, when addressing potentially active bottlenecks. A quantitative model of the VSL impact is proposed that allows for VSL to be incorporated in a macroscopic second-order traffic flow model as an additional control component. The integrated motorway network traffic control problem involving ramp metering and VSL control measures is formulated as a constrained discrete-time optimal control problem and is solved efficiently even for large-scale networks by a suitable feasible direction algorithm. An illustrative example of a hypothetical motorway stretch is investigated under different control scenarios, and it is shown that traffic flow efficiency can be substantially improved when VSL control measures are used, particularly in integration with coordinated ramp metering.

Second-order optic flow processing in amblyopia

Second-order optic flow processing in amblyopia

Comparative Study of Continuous and Pulsatile Left Ventricular Assist Devices on Hemodynamics of a Pediatric End-to-Side Anastomotic Graft.

Although there are many studies that focus on understanding the consequence of pumping mode (continuous vs. pulsatile) associated with ventricular assist devices (VADs) on pediatric vascular pulsatility, the impact on local hemodynamics has been largely ignored. Hence, we compare not only the hemodynamic parameters indicative of pulsatility but also the local flow fields in the aorta and the great vessels originating from the aortic arch. A physiologic graft anastomotic model is constructed based on a pediatric, patient specific, aorta with a graft attached on the ascending aorta. The flow is simulated using a previously validated second-order accurate Navier-Stokes flow solver based upon a finite volume approach. The major findings are: (1) pulsatile support provides a greater degree of vascular pulsatility when compared to continuous support, which, however, is still 20% less than pulsatility in the healthy aorta; (2) pulsatile support increases the flow in the great vessels, while continuous support decreases it; (3) complete VAD support results in turbulence in the aorta, with maximum principal Reynolds stresses for pulsatile support and continuous support of 7081 and 249 dyn/cm2, respectively; (4) complete pulsatile support results in a significant increase in predicted hemolysis in the aorta; and (5) pulsatile support causes both higher time-averaged wall shear stresses (WSS) and oscillatory shear indices (OSI) in the aorta than does continuous support. These findings will help to identify the risk of graft failure for pediatric patients with pulsatile and continuous VADs.

Stochastic characterization of the Montalto Uffugo research site (Italy) by geostatistical inversion of moment equations of groundwater flow

Summary We assess the applicability and performance of a methodology of inverting stochastic mean groundwater flow equations to characterize the spatial variability of (natural) log-transmissivity (Y) of a heterogeneous aquifer. The methodology, originally proposed by Hernandez et al., 2003 , Hernandez et al., 2006 , relies on a nonlinear geostatistical inverse algorithm for recursive approximations of steady-state (ensemble) mean groundwater flow that allows estimating jointly the spatial variability of Y, the underlying variogram parameters, and the variance–covariance of the estimates. Estimates of prediction errors of hydraulic heads and fluxes are then calculated a posteriori, upon solving equations satisfied by the corresponding covariances. Here, we extend the methodology to quasi-steady state flow conditions and present its first field application by using information collected during a pumping test performed at the Montalto Uffugo research site (Italy). Log-transmissivity is parameterized geostatistically on the basis of an available measured value and a set of unknown values at discrete pilot points. Best estimates of Y at the measurement location and at the pilot points are obtained by a maximum likelihood fit of computed and measured heads. These posterior estimates are then projected onto the computational grid by kriging. Information on head drawdowns is provided through self-potential signals recorded by 47 surface electrodes during the test. The maximum likelihood-based objective function includes a regularization term reflecting prior information about Y. The relative weight assigned to this term is evaluated separately from other model parameters to avoid bias and instability. We explore the effectiveness of both a zero- and a second-order closure of the mean flow equation at providing a proper geostatistical characterization of the log-transmissivity field. The parameters of the variogram of Y are estimated a posteriori using formal model selection criteria. The adoption of a second-order mean flow model renders the sharpest definition of the regularization term and of the Y variogram parameters.

Particle dispersion by random waves in rotating shallow water

We present a theoretical and numerical study of wave-induced particle dispersion due to random waves in the rotating shallow-water system, as part of an ongoing study of particle dispersion in the ocean. Specifically, the effective particle diffusivities in the sense of Taylor (Proc. Lond. Math. Soc., vol. 20, 1921, p. 196) are computed for a small-amplitude wave field modelled as a stationary homogeneous isotropic Gaussian random field whose frequency spectrum is bounded away from zero. In this case, the leading-order diffusivity depends crucially on the nonlinear, second-order corrections to the linear velocity field, which can be computed using the methods of wave–mean interaction theory. A closed-form analytic expression for the effective diffusivity is derived and carefully tested against numerical Monte Carlo simulations. The main conclusions are that Coriolis forces in shallow water invariably decrease the effective particle diffusivity and that there is a peculiar choking effect for the second-order particle flow in the limit of strong rotation.

Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion

Direct numerical simulations (DNS) of the wake of a circular disk placed normal to a uniform flow show that, as the Reynolds number is increased, the flow undergoes a sequence of successive bifurcations, each state being characterized by specific time and space symmetry breaking or recovering (Fabre, Auguste & Magnaudet, Phys. Fluids, vol. 20 (5), 2008, p. 1). To explain this bifurcation scenario, we investigate the stability of the axisymmetric steady wake in the framework of the global stability theory. Both the direct and adjoint eigenvalue problems are solved. The threshold Reynolds numbers Re and characteristics of the destabilizing modes agree with the study of Natarajan & Acrivos (J. Fluid Mech., vol. 254, 1993, p. 323): the first destabilization occurs for a stationary mode of azimuthal wavenumber m = 1 at RecA = 116.9, and the second destabilization of the axisymmetric flow occurs for two oscillating modes of azimuthal wavenumbers m ± 1 at RecB = 125.3. Since these critical Reynolds numbers are close to one another, we use a multiple time scale expansion to compute analytically the leading-order equations that describe the nonlinear interaction of these three leading eigenmodes. This set of equations is given by imposing, at third order in the expansion, a Fredholm alternative to avoid any secular term. It turns out to be identical to the normal form predicted by symmetry arguments. Though, all coefficients of the normal form are here analytically computed as the scalar product of an adjoint global mode with a resonant third-order forcing term, arising from the second-order base flow modification and harmonics generation. We show that all nonlinear interactions between modes take place in the recirculation bubble, as the contribution to the scalar product of regions located outside the recirculation bubble is zero. The normal form accurately predicts the sequence of bifurcations, the associated thresholds and symmetry properties observed in the DNS calculations.

Reply to comment on “On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models” by H.M. Zhang

This contribution replies to Zhang’s comment on the manuscript “On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models” published in Eur. Phys. J. B. The author clarifies several points and agrees that suitable experiments can be a way of deciding the controversy on the existence of propagation speeds faster than the speed of vehicles. Moreover, he proposes to aim at an integrated theory, which is both, theoretically consistent and practically relevant.

Comment on “On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models" by D. Helbing and A.F. Johansson

There has been a debate in the transportation research community over the validity of a class of traffic flow models. These models have the peculiar property that one of its characteristic speeds is faster than that of vehicular traffic. This note attempts to provide an overview of the diverging views on how to interpret this property, and specific comments on the interpretation of Helbing and Johansson [On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models, Eur. Phys. J. B (2009), DOI: 10.1140/epjb/e2009-00182-7]. We showed that having faster-than-traffic characteristics does produce counterintuitive predictions, and they cannot be explained away by a linear stability analysis. As such, the existence of such characteristics must be justified by the physics of traffic, and verified through empirical observations.

On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models

Daganzo’s criticisms of second-order fluid approximations of traffic flow [C. Daganzo, Transpn. Res. B. 29, 277 (1995)] and Aw and Rascle’s proposal how to overcome them [A. Aw, M. Rascle, SIAM J. Appl. Math. 60, 916 (2000)] have stimulated an intensive scientific activity in the field of traffic modeling. Here, we will revisit their arguments and the interpretations behind them. We will start by analyzing the linear stability of traffic models, which is a widely established approach to study the ability of traffic models to describe emergent traffic jams. Besides deriving a collection of useful formulas for stability analyses, the main attention is put on the characteristic speeds, which are related to the group velocities of the linearized model equations. Most macroscopic traffic models with a dynamic velocity equation appear to predict two characteristic speeds, one of which is faster than the average velocity. This has been claimed to constitute a theoretical inconsistency. We will carefully discuss arguments for and against this view. In particular, we will shed some new light on the problem by comparing Payne’s macroscopic traffic model with the Aw-Rascle model and macroscopic with microscopic traffic models.

The second-order renormalization group flow for nonlinear sigma models in two dimensions

We show that for two-dimensional manifolds M with negative Euler characteristics, there exist subsets of the space of smooth Riemannian metrics which are invariant and either parabolic or backwards-parabolic for the second-order RG flow. We also show that solutions exist globally on these sets. Finally, we establish the existence of an eternal solution that has both a UV and IR limit, and passes through regions where the flow is parabolic and backwards-parabolic.

Three-particle cumulant study of conical emission

We discuss the sensitivity of the three-particle azimuthal cumulant method for a search and study of conical emission in central relativistic $A+A$ collisions. Our study is based on a multicomponent Monte Carlo model that includes flow background, Gaussian monojets, jet flow, and Gaussian conical signals. We find the observation of conical emission is hindered by the presence of flow harmonics of fourth order (${v}_{4}$) but remains feasible even in the presence of a substantial background. We consider the use of probability cumulants for the suppression of second-order flow harmonics. We find that although probability cumulant significantly reduce ${v}_{2}^{2}$ contributions, they also complicate the cumulant of jets and conical emission. The use of probability cumulants is therefore not particularly advantageous in searches for conical emission. We find the sensitivity of the (density) cumulant method depends inextricably on strengths of ${v}_{2}$, ${v}_{4}$, background, and non-Poisson character of particle production. It thus cannot be expressed in a simple form and without specific assumptions about the values of these parameters.

Microtube gas flows with second-order slip flow and temperature jump boundary conditions

Second-order slip flow and temperature jump boundary conditions are applied to solve the momentum and energy equations in a microtube for an isoflux thermal boundary condition. The flow is assumed to be hydrodynamically fully developed, and the thermal field is either fully developed or developing from the tube entrance. In general, second-order boundary conditions assuming an effective mean free path model predict a lower slip velocity than a first-order model assuming a hard sphere mean free path model. Heat transfer effects associated with rarefied flow are reduced for the second-order model. The effect of the second-order terms is most significant at the upper limit of the slip regime. For airflow at standard conditions, the maximum second-order change to the Nusselt number is on the order of 15%. The second-order effect is also more significant in the entrance region of the tube. Nusselt numbers are found to increase relative to their no-slip values when temperature jump effects are small. In cases where slip and temperature jump effects are of the same order, or where temperature jump effects dominate, the Nusselt number decreases when compared to traditional no-slip conditions.

Two spheres in a free stream of a second-order fluid

The forces acting on two fixed spheres in a second-order uniform flow are investigated. When α1+α2=0, where α1 and α2 are fluid parameters related to the first and second normal stress coefficients, the velocity field for a second-order fluid is the same as the one predicted by the Stokes equations while the pressure is modified. The Stokes solutions given by Stimson and Jeffery [Proc. R. Soc. London, Ser. A 111, 110 (1926)] for the case when the flow direction is along the line of centers and Goldman et al. [Chem. Eng. Sci. 21, 1151 (1966)] for the case when the flow direction is perpendicular to the line of centers are utilized and the stresses and the forces acting on the particles in a second-order fluid are calculated. For flow along the line of centers or perpendicular to it, the net force is in the direction that tends to decrease the particle separation distance. For the case of flow at arbitrary angle, unequal forces are applied to the spheres perpendicularly to the line of centers. These forces result in a change of orientation of the sedimenting spheres until the line of centers aligns with the flow direction. In addition, the potential flow of a second-order fluid past two fixed spheres in a uniform flow is investigated. The normal stress at the surface of each sphere is calculated and the viscoelastic effects on the normal stress for different separation distances are analyzed. The contribution of the potential flow of a second-order fluid to the force applied to the particles is an attractive force. Our explanations of the aggregation of particles in viscoelastic fluids rest on three pillars; the first is a viscoelastic “pressure” generated by normal stresses due to shear. Second, the total time derivative of the pressure is an important factor in the forces applied to moving particles. The third is associated with a change in the normal stress at points of stagnation which is a purely extensional effect unrelated to shearing.

Numerical study of heat transfer of a third grade viscoelastic fluid in non-Darcy porous media with thermophysical effects

A numerical solution is presented for the natural convective dissipative heat transfer of an incompressible, third grade, non-Newtonian fluid flowing past an infinite porous plate embedded in a Darcy–Forchheimer porous medium. The mathematical model is developed in an (x,y) coordinate system. Using a set of transformations, the momentum equation is rendered one-dimensional and a partly linearized heat conservation equation is derived. The viscoelastic formulation presented by Akyildiz (2001 Int. J. Non-Linear Mechanics36 349–52) is adopted, which generates lateral mass and viscoelastic terms in the heat conservation equation, as well as in the momentum equation. A number of special cases of the general transformed model are discussed. A finite element method is implemented to solve, with appropriate boundary conditions, the coupled third-order, second degree ordinary differential equation for momentum and the second-order, fourth degree heat conservation equation. We study the influence of the third grade viscoelastic parameter (β3), Darcian parameter (inversely proportional to permeability (kp)), the Forchheimer inertial parameter (b), transpiration velocity (Vo), the transpiration parameter in the heat equation (R) and the thermal conductivity parameter (S) on momentum and heat transfer. Additionally, we study the influence of the Forchheimer inertial parameter (b) on second-order viscoelastic non-Darcy free convection flow and also the effects of the third grade parameter (β3) on Darcian free convection. Velocities increase with rising permeability (Darcian parameter) for both second and third grade viscoelastic free convection regimes and decrease with rising Forchheimer parameter. The effects of the other parameters are described at length. The flow scenario is important in chemical engineering processes.

Integrated Ramp Metering and Variable Speed Limit Control of Motorway Traffic Flow

The impact of variable speed limits (VSL) on the aggregate traffic flow behaviour is reflected in the quantitative model proposed in this paper. VSL are incorporated in a general second-order traffic flow model as an additional control component. The integrated motorway network traffic control problem is formulated as a constrained discrete-time optimal control problem which is solved very efficiently even for large-scale networks by a suitable feasible-direction algorithm. An illustrative example is presented under different control scenarios and it is shown that traffic flow efficiency can be substantially improved when VSL control measures are used, particularly in integration with coordinated ramp metering.

Second-Order Optic Flow Deficits in Amblyopia

Amblyopic observers show deficits for global motion discrimination that cannot be accounted for by their contrast sensitivity impairment. The processing of first- and second-order translational global motion is deficient, as is the processing of first-order optic flow, suggesting that cortical function in extrastriate areas is impaired. The authors sought to determine whether amblyopes show impairment in the processing of optic flow defined by second-order motion, whether these deficits are comparable in the two eyes, and whether these deficits are correlated with first-order deficits. Eight amblyopic subjects (three strabismic, three strabismic-anisometropic, one anisometropic, one deprivation; mean age, 29 years) were tested. The authors used random dot kinematograms in which the dots were luminance or contrast modulations of background noise. The global pattern of dot motion within the stimulus area was translational, radial, or rotational. Coherence thresholds for direction discrimination were obtained across a range of dot modulation depths, allowing the separation of contrast and motion deficits. The present study showed that deficits in second-order optic flow processing were equivalent to those for first-order stimuli and that these were unrelated to the extent of the amblyopic contrast sensitivity deficit and were comparable in both eyes. Radial optic flow was more affected than rotational optic flow. Global motion impairment appeared to have a high-level binocular locus and was independent of the depth of the contrast deficit. Results also support the idea that global motion and optic flow processing are form-cue invariant.

Fast parametrically excited van der Pol oscillator with time delay state feedback

We investigate the effect of a fast vertical parametric excitation on self-excited vibrations in a delayed van der Pol oscillator. We use the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic in the vicinity of the trivial equilibrium. Then, we apply the multiple scales method on this slow dynamic to derive a second-order slow flow system describing the modulation of slow dynamic. In particular we analyze the slow flow to obtain the effect of a fast excitation on the regions in parameter space where self-excited vibrations can be eliminated. We have shown that in the case where the time delay and the feedback gains are imposed, fast vertical parametric excitation can be an alternative to suppress undesirable self-excited vibrations in a delayed van der Pol oscillator.

Large-Eddy Simulation of the Stable Atmospheric Boundary Layer using Dynamic Models with Different Averaging Schemes

Large-eddy simulation (LES) of a stable atmospheric boundary layer is performed using recently developed dynamic subgrid-scale (SGS) models. These models not only calculate the Smagorinsky coefficient and SGS Prandtl number dynamically based on the smallest resolved motions in the flow, they also allow for scale dependence of those coefficients. This dynamic calculation requires statistical averaging for numerical stability. Here, we evaluate three commonly used averaging schemes in stable atmospheric boundary-layer simulations: averaging over horizontal planes, over adjacent grid points, and following fluid particle trajectories. Particular attention is focused on assessing the effect of the different averaging methods on resolved flow statistics and SGS model coefficients. Our results indicate that averaging schemes that allow the coefficients to fluctuate locally give results that are in better agreement with boundary-layer similarity theory and previous LES studies. Even among models that are local, the averaging method is found to affect model coefficient probability density function distributions and turbulent spectra of the resolved velocity and temperature fields. Overall, averaging along fluid pathlines is found to produce the best combination of self consistent model coefficients, first- and second-order flow statistics and insensitivity to grid resolution.