The effect of canopy roughness density on the constitutive components of the dispersive stresses

How to represent the effects of variable canopy morphology on turbulence remains a fundamental challenge yet to be confronted. Planar averaging over some minimal area can be applied to average-out this sort of spatial variability in the time-averaged mean momentum balance. Because of the multiply connected air-spaces, spatial averaging gives rise to covariance or dispersive stress terms that are produced by the spatial correlations of the time-averaged quantities. These terms are “unclosed” and require parameterization, which to date remains lacking due to the absence of data. Here, flume experiments were conducted to quantify the magnitude and sign of the dispersive stresses for a cylindric canopy where the rod density was varied but the individual rod dimensions (rod height h _c and rod diameter d _r) remained the same. Quadrant analysis was used to explore the genesis of their spatial coherency inside the canopy for a wide range of rod densities. When compared to the conventional turbulent stresses, these dispersive stresses can be significant in the lowest layers of sparse canopies. For dense canopies, the dispersive terms remain negligible when compared to the conventional momentum fluxes at all the canopy levels consistent with previous experiments in vegetated and urban canopies. It was also shown that the spatial locations contributing most to the dispersive terms were in the immediate vicinity downstream of the rods. In the deeper layers of sparse canopies, these positions contributed large and negative stresses, but in the upper levels of the canopy, they contributed large but positive stresses. Because the longitudinal velocity spatial perturbation behind the rods is negative, the switch in sign in these stresses was connected with the sign of the vertical velocity spatial perturbation Simplified scaling arguments, using a reduced mean continuity equation and the vertical mean momentum balance for the flow field near the rods, offer clues as to why in much of the lower canopy levels (about 0.75 h _c ) while in the upper canopy levels.     

Reynolds stress field in a turbulent wall jet induced by a streamwise vortex with periodic perturbation

An experimental study on the Reynolds stress tensor was conducted in the three-dimensional flow in the plane turbulent wall jet induced by an isolated streamwise vortex generated by the half-delta wing mounted on the wall. Oscillation of the angle of attack of the wing induced a periodic perturbation in the strength of the streamwise vortex. Analysis by triple velocity decomposition and phase averaging shows that the oscillation induces periodic variations in the strength, radius, and position of the streamwise vortex center. The effect of periodic perturbation manifests itself in the magnitude of the Reynolds stress components and Simulations prove that the periodic variations in the strength, radius, and position of the vortex center can generate an apparent shear stress, denoted herein as      

On the Non-Isentropic Sound Waves Propagation in a Cylindrical Tube Filled with Saturated Porous Media

A rigid frame, cylindrical capillary theory of sound propagation in porous media that includes the nonlinear effects of the Forchheimer type is laid out by using variational solutions. It is shown that the five main parameters governing the propagation of sound waves in a fluid contained in rigid cylindrical tubes filled with a saturated porous media are: the shear wave number, , the reduced frequency parameter, , the porosity, ε, Darcy number, , and Forchheimer number, . The manner in which the flow influences the attenuation and the phase velocities of the forward and backward propagating non-isentropic acoustic waves is deduced. It is found that the inclusion of the solid matrix increases wave’s attenuations and phase velocities for both forward and backward sound waves, while increasing the porosity and the reduced frequency number decreased attenuation and increased phase velocities. The effect of the steady flow is found to decrease the attenuation and phase velocities for forward sound waves, and enhance them for the backward sound waves. This work is done during a sabbatical leave year granted form the University of Jordan to Dr. Hamzeh Duwairi for the academic year 2007/2008 at the German Jordanian University.     

Vector potential–vorticity relationship for the Stokes flows: application to the Stokes eigenmodes in 2D/3D closed domain

The unsteady dynamics of the Stokes flows, where , is shown to verify the vector potential–vorticity ( ) correlation , where the field is the pressure-gradient vector potential defined by . This correlation is analyzed for the Stokes eigenmodes, , subjected to no-slip boundary conditions on any two-dimensional (2D) closed contour or three-dimensional (3D) surface. It is established that an asymptotic linear relationship appears, verified in the core part of the domain, between the vector potential and vorticity, , where is a constant offset field, possibly zero.     

Flow Visualization of Superbursts and of the Log-Layer in a DNS at $$ \operatorname{Re} _{\tau } = 950 $$

This paper uses direct numerical simulations (DNS) of turbulent flow in a channel at (Del álamo, Jiménez, Zandonade, Moser J Fluid Mech 500:135–144, 2004) to provide a picture of the turbulent structures making large contributions to the Reynolds shear stress. Considerable work of this type has been done for the viscous wall region at smaller , for which a log-layer does not exist. Recent PIV measurements of turbulent velocity fluctuations in a plane parallel to the direction of flow have emphasized the dominant contribution of large scale structures in the outer flow. This prompted Hanratty and Papavassiliou (The role of wall vortices in producing turbulence. In: Panton, R.L. (ed) Self-sustaining Mechanism of Wall Turbulence. Computational Mechanics Publications, Southampton, pp. 83–108, 1997) to use DNS at to examine these structures in a plane perpendicular to the direction of flow. They identified plumes which extend from the wall to the center of a channel. The data at are used to explore these results further, to examine the structure of the log-layer, and to test present notions about the viscous wall layer.     

Vortex formation process in gravity-driven starting jets

Experiments were conducted to investigate the vortex formation process in gravity-driven starting jets at three Reynolds numbers ( 3,528 and 4,716, where D is the nozzle diameter, the average discharging velocity and ν the kinematic viscosity). Planar laser induced fluorescence (PLIF) was used to visualize the flow while particle image velocimetry (PIV) was used to quantify the 2D velocity and vorticity fields. Vortex leapfrogging was found to appear at Re = 2,358, while pinch-off for the leading vortex was observed in the other two cases. Conditions for the vortex pinch-off appeared to be different from those found in the piston-driven starting jets. Although the leading vortex attained the maximum circulation and energy level very quickly after the jet was initiated, its detachment from the main stem may not be achieved necessarily. Attempt had been made to elaborate the different flow characteristics for respective cases from the related distribution of circulation and energy level via the measured velocity field by PIV.     

Resolvent Estimates and Maximal Regularity in Weighted L q -spaces of the Stokes Operator in an Infinite Cylinder

Let be an infinite cylinder of , n ≥ 3, with a bounded cross-section of C ^1,1-class. We study resolvent estimates and maximal regularity of the Stokes operator in for 1 < q, r < ∞ and for arbitrary Muckenhoupt weights ω ∈ A _r with respect to x′ ∈ Σ. The proofs use an operator-valued Fourier multiplier theorem and techniques of unconditional Schauder decompositions based on the -boundedness of the family of solution operators for a system in Σ parametrized by the phase variable of the one-dimensional partial Fourier transform. Supported by the Gottlieb Daimler- und Karl Benz-Stiftung, grant no. S025/02-10/03.     

The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation

We study the limit of the hyperbolic–parabolic approximation

Two-Phase Inertial Flow in Homogeneous Porous Media: A Theoretical Derivation of a Macroscopic Model

The purpose of this article is to derive a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian fluid flow through homogenous porous media. Starting from the continuity and Navier–Stokes equations in each phase β and γ, the method of volume averaging is employed subjected to constraints that are explicitly provided to obtain the macroscopic mass and momentum balance equations. These constraints are on the length- and time-scales, as well as, on some quantities involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The resulting macroscopic momentum equation relates the phase-averaged pressure gradient to the filtration or Darcy velocity in a coupled nonlinear form explicitly given by

The Steady Motion of a Navier–Stokes Liquid Around a Rigid Body

Let be a body moving by prescribed rigid motion in a Navier–Stokes liquid that fills the whole space and is subject to given boundary conditions and body force. Under the assumptions that, with respect to a frame , attached to , these data are time independent, and that their magnitude is not “too large”, we show the existence of one and only one corresponding steady motion of , with respect to , such that the velocity field, at the generic point x in space, decays like |x|⁻¹. These solutions are “physically reasonable” in the sense of FINN [10]. In particular, they are unique and satisfy the energy equation. Among other things, this result is relevant in engineering applications involving orientation of particles in viscous liquid [14].     

Momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel

Unsteady momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel is numerically investigated using FLUENT for the ranges of Reynolds numbers as 10≤Re≤500, of the blockage ratio as 0.1≤β≤0.4, and of the gap ratio as 0.125≤γ≤1 for a constant value of the Prandtl number of 0.744. The transition of the flow from steady to unsteady (characterized by critical Re) is determined as a function of γ and β. The effect of γ on the mean drag and lift coefficients, Strouhal number (St), and Nusselt number (Nu _w ) is studied. Critical Re was found to increase with decreasing γ for all values of β. and St were found to increase with decreasing values of γ for fixed β and Re. The effect of decrease in γ on was found to be negligible for all blockage ratios investigated.     

Realization of Period Maps of Planar Hamiltonian Systems

We consider the set of 2π-periodic solutions of the ordinary differential equation u′′ + g(u) = 0 for a nonlinearity , satisfying a dissipative condition of the form for , and under the generic assumption that the potential G, given by , is a Morse function. Under these assumptions, we characterize the period maps realizable by planar Hamiltonian systems of the form . Considering the Morse type of G, the set of periodic orbits in the phase space is decomposed into disks and annular regions. Then, the realizable period maps are described in terms of sets of sequences of positive integers corresponding to the lap numbers of the 2π-periodic solutions. This leads to a characterization of the classes of Morse–Smale attractors that are realizable by dissipative semilinear parabolic equations of the form defined on the circle, .      

A Jordan Curve Spanned by a Complete Minimal Surface

In this paper we construct complete (conformal) minimal immersions which admit continuous extensions to the closed disk, . Moreover, is a homeomorphism and is a (non-rectifiable) Jordan curve with Hausdorff dimension 1. It turns out that the set of Jordan curves constructed by the above procedure is dense in the space of Jordan curves of with the Hausdorff metric.     

Evolution of viscosities and morphology for the two-phase system polyethylene oxide/poly(dimethylsiloxane)

Shear viscosities and oscillatory viscosities were measured for the two-phase system polyethylene oxide/poly(dimethylsiloxane) at 70 °C as a function of composition. This blend exhibits the usual droplet/matrix structures in the vicinity of the pure components and a region of co-continuity within which two droplet/matrix structures coexist. A stepwise reduction in the shear rate, , leads to a rapid increase in viscosity followed by a much slower exponential decay; plots of the corresponding rate constants as a function of composition exhibit two discontinuities marking the boundaries of co-continuity; a similar dependence is obtained for the time independent final viscosities . Keeping the blend composition constant and determining as a function of yields a curve that passes a distinct maximum, where the viscosities are very close to that of the less viscous pure component on both ends of this dependence. The dynamic mechanical measurements of the blends yield at low frequencies storage moduli G′ that are orders of magnitude larger than that of the components because of the deformation of the interfaces. At high frequencies, the loss moduli G″ reflect the increasing alignment of the drops suspended in the matrix phases. The composition dependencies of G′ and of the complex viscosities can again be used to determine the limits of co-continuity.     

FPIV study of gas entrainment by a hollow cone spray submitted to variable density

The gas entrainment in a hollow cone spray submitted to variable density is studied experimentally in order to better understand the effect on mixture formation. Particle image velocimetry on fluorescent tracers, associated with a specific processing of the instantaneous velocity fields have been applied to obtain measurement in the close vicinity of the spray edge. In the “quasi-steady” region of the spray, important effect of the ambient density on the mass flow rate of entrained gas have been pointed out. The axial evolution of is in good agreement with an integral model that takes the momentum exchange between phases into account.     

Non-autonomous 2D Navier–Stokes System with Singularly Oscillating External Force and its Global Attractor

We study the global attractor of the non-autonomous 2D Navier–Stokes (N.–S.) system with singularly oscillating external force of the form . If the functions g ₀(x, t) and g ₁ (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor is bounded in the space H, however, its norm may be unbounded as since the magnitude of the external force is growing. Assuming that the function g ₁ (z, t) has a divergence representation of the form where the functions (see Section 3), we prove that the global attractors of the N.–S. equations are uniformly bounded with respect to for all . We also consider the “limiting” 2D N.–S. system with external force g ₀(x, t). We have found an estimate for the deviation of a solution of the original N.–S. system from a solution u ⁰(x, t) of the “limiting” N.–S. system with the same initial data. If the function g ₁ (z, t) admits the divergence representation, the functions g ₀(x, t) and g ₁ (z, t) are translation compact in the corresponding spaces, and , then we prove that the global attractors converges to the global attractor of the “limiting” system as in the norm of H. In the last section, we present an estimate for the Hausdorff deviation of from of the form: in the case, when the global attractor is exponential (the Grashof number of the “limiting” 2D N.–S. system is small).      

Simultaneous measurement of turbulent velocity field and surface wave amplitude in the initial stage of an open-channel flow by PIV

Particle image velocimetry (PIV) was employed to measure the two components of the turbulent velocity field in the initial stage of an open-channel flow in a streamwise-wall-normal plane, and the free-surface level was discriminated from the PIV image. The details of this technique was described and demonstrated by showing the instantaneous velocity field together with the free-surface shape, statistics of velocity field, and the wave-turbulence interaction terms. Preliminary experimental results showed that the turbulence intensity of the streamwise velocity fluctuations (u′) decreased, whereas that of the wall-normal velocity (v′) increased near the bottom wall with downstream distance in the initial stage of an open-channel flow; (g is the fluctuation of free-surface level) had a negative value, had a positive value near the free surface, and the surface-wave-affected depth deepened with downstream distance.     

Full Measure Reducibility for Generic One-parameter Family of Quasi-periodic Linear Systems

Let be the set of m × m matrices A(λ) depending analytically on a parameter λ in a closed interval . Consider one-parameter families of quasi-periodic linear differential equations: , where is analytic and sufficiently small. We prove that there is an open and dense set in , such that for each the equation can be reduced to an equation with constant coefficients by a quasi-periodic linear transformation for almost all in Lebesgue measure sense provided that g is sufficiently small. The result gives an affirmative answer to a conjecture of Eliasson (In: Proceeding of Symposia in Pure Mathematics). Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

Rotating Navier-Stokes Equations in $${\mathbb R}^{3}_{+}$$ with Initial Data Nondecreasing at Infinity: The Ekman Boundary Layer Problem

We prove time local existence and uniqueness of solutions to a boundary layer problem in a rotating frame around the stationary solution called the Ekman spiral. We choose initial data in the vector-valued homogeneous Besov space for 2 <  p <  ∞. Here the L ^p -integrability is imposed in the normal direction, while we may have no decay in tangential components, since the Besov space contains nondecaying functions such as almost periodic functions. A crucial ingredient is theory for vector-valued homogeneous Besov spaces. For instance we provide and apply an operator-valued bounded H ^∞-calculus for the Laplacian in for a general Banach space .