Two-dimensional turbulence on a sphere

Following Fjørtoft (Tellus, vol. 5, 1953, pp. 225–230) we undertake a spectral analysis of a non-divergent flow on a sphere. It is shown that the spherical harmonic energy spectrum is invariant under rotations of the polar axis of the spherical harmonic system and argued that a constraint of isotropy would not simplify the analysis but only exclude low-order modes. The spectral energy equation is derived and it is shown that the viscous term has a slightly different form than given in previous studies. The relations involving energy transfer within a triad of modes, which Fjørtoft (Tellus, vol. 5, 1953, pp. 225–230) derived under the condition that energy transfer is restricted to three modes, are derived under general conditions. These relations show that there are two types of interaction within a triad. The first type is where the middle mode acts as a source for the two other modes and the second type is where it acts as a sink. The inequality indicating cascade directions which was derived by Gkioulekas & Tung (J. Fluid Mech., vol. 576, 2007, pp. 173–189) in Fourier space under the assumptions of narrow band forcing and stationarity is derived in spherical harmonic space under the assumption of dominance of first type interactions. The double cascade theory of Kraichnan (Phys. Fluids, vol. 10, 1967, pp. 1417–1423) is discussed in the light of the derived equations and it is hypothesised that in flows with limited scale separation the two cascades may, to a large extent, be produced by the same triad interactions. Finally, we conclude that the spherical geometry is the optimal test ground for exploration of two-dimensional turbulence by means of simulations.

Effect of Dissipation on the Moonpool-Javelin Wave Energy Converter

In this work, the hydrodynamic performance of a novel wave energy converter (WEC) configuration which combines a moonpool platform and a javelin floating buoy, called the moonpool–javelin wave energy converter (MJWEC), was studied by semianalytical, computational fluid dynamics (CFD), and experimental methods. The viscous term is added to the potential flow solver to obtain the hydrodynamic coefficients. The wave force, the added mass, the radiation damping, the wave capture, and the energy efficiency of the configuration were assessed, in the frequency and time domains, by a semianalytical method. The CFD method results and the semianalytical results were compared for the time domain by introducing nonlinear power take-off (PTO) damping; additionally, the viscous dissipation coefficients under potential flow could be confirmed. Finally, a 1:10 scale model was physically tested to validate the numerical model and further prove the feasibility of the proposed system.

On the Viscous Cahn-Hilliard-Oono System with Chemotaxis and Singular Potential

We analyze a diffuse interface model that couples a viscous Cahn-Hilliard equation for the phase variable with a diffusion-reaction equation for the nutrient concentration. The system under consideration also takes into account some important mechanisms like chemotaxis, active transport as well as nonlocal interaction of Oono’s type. When the spatial dimension is three, we prove the existence and uniqueness of global weak solutions to the model with singular potentials including the physically relevant logarithmic potential. Then we obtain some regularity properties of the weak solutions when t>0. In particular, with the aid of the viscous term, we prove the so-called instantaneous separation property of the phase variable such that it stays away from the pure states ±1 as long as t>0. Furthermore, we study long-time behavior of the system, by proving the existence of a global attractor and characterizing its ω-limit set.

The Viscosity Effect on Velocity of a Macroscopic Vehicular Traffic Model

Traffic in Mexico City poses a serious problem of vehicle saturation that causes a decrease in speed and increased transport time in the streets that suffer mobility collapses. A macroscopic model of vehicular traffic is used to show the effect of viscosity on the vehicular variables (speed and vehicle density), applied to two avenues in Mexico City, is studied. The input parameters were calculated following the Greenberg model. As the original model presents numerical divergences, the two assumptions corresponding to conservation of the vehicle’s mass and the viscous term are modified. The results suggest that the viscosity depends on time and that it can be adapted to recommend modifications in urban mobility parameters, or even to implement the public planning policies in construction of infrastructure for urban transport, to make vehicle flow more efficient.

Two Dimensional Analysis of Hybrid Spectral/Finite Difference Schemes for Linearized Compressible Navier–Stokes Equations

This study seeks to compare different combinations of spatial dicretization methods under a coupled spatial temporal framework in two dimensional wavenumber space. The aim is to understand the effect of dispersion and dissipation on both the convection and diffusion terms found in the two dimensional linearized compressible Navier–Stokes Equations (LCNSE) when a hybrid finite difference/Fourier spectral scheme is used in the x and y directions. In two dimensional wavespace, the spectral resolution becomes a function of both the wavenumber and the wave propagation angle, the orientation of the wave front with respect to the grid. At sufficiently low CFL number where temporal discretization effects can be neglected, we show that a hybrid finite difference/Fourier spectral schemes is more accurate than a full finite difference method for the two dimensional advection equation, but that this is not so in the case of the LCNSE. Group velocities, phase velocities as well as numerical amplification factor were used to quantify the numerical anisotropy of the dispersion and dissipation properties. Unlike the advection equation, the dispersion relation representing the acoustic modes of the LCNSE contains an acoustic terms in addition to its advection and viscous terms. This makes the group velocity in each spatial direction a function of the wavenumber in both spatial directions. This can lead to conditions for which a hybrid Fourier spectral/finite difference method can become less or more accurate than a full finite difference method. To better understand the comparison of the dispersion properties between a hybrid and full FD scheme, the integrated sum of the error between the numerical group velocity $$V^{*}_{grp,full}$$ V g r p , f u l l ∗ and the exact solution across all wavenumbers for a range of wave propagation angle is examined. In the comparison between a hybrid and full FD discretization schemes, the fourth order central (CDS4), fourth order dispersion relation preserving (DRP4) and sixth order central compact (CCOM6) schemes share the same characteristics. At low wave propagation angle, the integrated errors of the full FD and hybrid discretization schemes remain the same. At intermediate wave propagation angle, the integrated error of the full FD schemes become smaller than that of the hybrid scheme. At large wave propagation angle, the integrated error of the full FD schemes diverges while the integrated error of the hybrid discretization schemes converge to zero. At high reduced wavenumber and sufficiently low CFL number where temporal discretization error can be neglected, it was found that the numerical dissipation of the viscous term based on the CDS4, DRP4, CCOM6 and isotropy optimized CDS4 schemes ($$\hbox {CDS4}_{{opt}}$$ CDS4 opt ) schemes was lower than the actual physical dissipation, which is only a function of the cell Reynolds number. The wave propagation angle at which the numerical dissipation of the viscous term approaches its maximum occurs at $$\pi /4$$ π / 4 for the CDS4, DRP4, CCOM6 and $$\hbox {CDS4}_{{opt}}$$ CDS4 opt schemes.

Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation

<p style='text-indent:20px;'>We consider one-dimensional distributed optimal control problems with the state equation being given by the viscous Burgers equation. We discretize using a space-time discontinuous Galerkin approach. We use upwind flux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our findings by numerical results.</p>

Modelling and Analysis of Complex Viscous Fluid in Thin Elastic Tubes

Cardiovascular disease is a major threat to human health. The study on the pathogenesis and prevention of cardiovascular disease has received special attention. In this paper, we have contributed to the derivation of a mathematical model for the nonlinear waves in an artery. From the Navier–Stokes equations and continuity equation, the vorticity equation satisfied by the blood flow is established. And based on the multiscale analysis and perturbation method, a new model of the Boussinesq equation with viscous term is derived to describe the propagation of a viscous fluid through a thin tube. In order to be more consistent with the flow of the fluid, the time-fractional Boussinesq equation with viscous term is deduced by employing the semi-inverse method and the fractional variational principle. Moreover, the approximate analytical solution of the fractional equation is obtained, and the effect of viscosity on the amplitude and width of the wave is studied. Finally, the effects of the fractional order parameters and vessel radius on blood flow volume are discussed and analyzed.