By using direct numerical simulations (DNS) at unprecedented resolution, we study turbulence under rotation in the presence of simultaneous direct and inverse cascades. The accumulation of energy at ...large scale leads to the formation of vertical coherent regions with high vorticity oriented along the rotation axis. By seeding the flow with millions of inertial particles, we quantify—for the first time—the effects of those coherent vertical structures on the preferential concentration of light and heavy particles. Furthermore, we quantitatively show that extreme fluctuations, leading to deviations from a normal-distributed statistics, result from the entangled interaction of the vertical structures with the turbulent background. Finally, we present the first-ever measurement of the relative importance between Stokes drag, Coriolis force, and centripetal force along the trajectories of inertial particles. We discover that vortical coherent structures lead to unexpected diffusion properties for heavy and light particles in the directions parallel and perpendicular to the rotation axis.
The hydrodynamic force exerted by a fluid on small isolated rigid spherical particles are usually well described by the Maxey–Riley (MR) equation. The most time-consuming contribution in the MR ...equation is the Basset history force which is a well-known problem for many-particle simulations in turbulence. In this paper a novel numerical approach is proposed for the computation of the Basset history force based on the use of exponential functions to approximate the tail of the Basset force kernel. Typically, this approach not only decreases the cpu time and memory requirements for the Basset force computation by more than an order of magnitude, but also increases the accuracy by an order of magnitude. The method has a temporal accuracy of O(Δt2) which is a substantial improvement compared to methods available in the literature. Furthermore, the method is partially implicit in order to increase stability of the computation. Traditional methods for the calculation of the Basset history force can influence statistical properties of the particles in isotropic turbulence, which is due to the error made by approximating the Basset force and the limited number of particles that can be tracked with classical methods. The new method turns out to provide more reliable statistical data.
We investigate the influence of shear on the gravitational settling of heavy inertial particles in homogeneous shear turbulence (HST). In addition to the well-known enhanced settling velocity, ...observed for heavy inertial particles in homogeneous isotropic turbulence (HIT), a horizontal drift velocity is also observed in the shearing direction due to the presence of a nonzero mean vorticity (introducing symmetry breaking due to the mean shear). This drift velocity is due to the combination of shear, gravity, and turbulence, and all three of these elements are needed for this effect to occur. We extend the mechanism responsible for the enhanced settling velocity in HIT to the case of HST. Two separate regimes are observed, characterized by positive or negative drift velocity, depending on the particle settling velocity.
In this paper a general theory for interpolation methods on a rectangular grid is introduced. By the use of this theory an efficient B-spline-based interpolation method for spectral codes is ...presented. The theory links the order of the interpolation method with its spectral properties. In this way many properties like order of continuity, order of convergence, and magnitude of errors can be explained. Furthermore, a fast implementation of the interpolation methods is given. We show that the B-spline-based interpolation method has several advantages compared to other methods. First, the order of continuity of the interpolated field is higher than for other methods. Second, only one FFT is needed, whereas, for example, Hermite interpolation needs multiple FFTs for computing the derivatives. Third, the interpolation error almost matches that of Hermite interpolation, a property not reached by other methods investigated.
An important aspect in numerical simulations of particle-laden turbulent flows is the interpolation of the flow field needed for the computation of the Lagrangian trajectories. The accuracy of the ...interpolation method has direct consequences for the acceleration spectrum of the fluid particles and is therefore also important for the correct evaluation of the hydrodynamic forces for almost neutrally buoyant particles, common in many environmental applications. In order to systematically choose the optimal tradeoff between interpolation accuracy and computational cost we focus on comparing errors: the interpolation error is compared with the discretization error of the flow field. In this way one can prevent unnecessary computations and still retain the accuracy of the turbulent flow simulation. From the analysis a practical method is proposed that enables direct estimation of the interpolation and discretization error from the energy spectrum. The theory is validated by means of direct numerical simulations (DNS) of homogeneous, isotropic turbulence using a spectral code, where the trajectories of fluid tracers are computed using several interpolation methods. We show that B-spline interpolation has the best accuracy given the computational cost. Finally, the optimal interpolation order for the different methods is shown as a function of the resolution of the DNS simulation.
The Stokes drag force and the gravity force are usually sufficient to describe the behavior of sub-Kolmogorov-size (or pointlike) heavy particles in turbulence, in particular when the ...particle-to-fluid density ratio ρ_{p}/ρ_{f}≳10^{3} (with ρ_{p} and ρ_{f} the particle and fluid density, respectively). This is, in general, not the case for smaller particle-to-fluid density ratios, in particular not for ρ_{p}/ρ_{f}≲10^{2}. In that case the pressure gradient force, added mass effects, and the Basset history force also play important roles. In this study we focus on the understanding of the role of these additional forces, all of hydrodynamic origin, in the settling of particles in turbulence. In order to qualitatively elucidate the complex dynamics of such particles in homogeneous isotropic turbulence, we first focus on the case of settling of such particles in the flow field of a single vortex. After having explored this simplified case we extend our analysis to homogeneous isotropic turbulence. In general, we found that the pressure gradient force leads to a decrease in the settling velocity. This can be qualitatively understood by the fact that this force prevents the particles from sweeping out of vortices, a mechanism known as preferential sweeping which causes enhanced settling. Additionally, we found that the Basset history force can both increase and decrease the enhanced settling, depending on the particle Stokes number. Finally, the role of the nonlinear Stokes drag has been explored, confirming that it affects settling of inertial particles in turbulence, but only in a limited way for the parameter settings used in this investigation.
An important aspect in numerical simulations of particle laden turbulent flows is the interpolation of the flow field. For the interpolation different approaches are used. Where some studies use low ...order linear interpolation others use high order spline methods. We compare several interpolation methods and conclude that interpolation based on B-spline functions has several advantages compared with traditional methods. First, B-spline interpolation can be executed very efficiently by optimal use of the pseudo-spectral code, only one FFT needs to be executed where Hermite spline needs multiple FFTs for computing the derivatives. Second, the smoothness of the interpolated field is higher than that of Hermite spline interpolation. Finally, the interpolation error almost matches the one of Hermite spline which is not reached by the other methods investigated. Further, we focus on estimating the interpolation error and compare it with the discretisation error of the flow field. In this way one can balance the errors in order to achieve an optimal result. Algorithms have been developed for the approximation of the interpolation error. As a spin-off of the theoretical analysis a practical method is proposed which enables direct estimation of the interpolation error from the energy spectrum, which may provide a quantitative indicator for this purpose.
By using direct numerical simulations (DNS) at unprecedented resolution we study turbulence under rotation in the presence of simultaneous direct and inverse cascades. The accumulation of energy at ...large scale leads to the formation of vertical coherent regions with high vorticity oriented along the rotation axis. By seeding the flow with millions of inertial particles, we quantify -for the first time- the effects of those coherent vertical structures on the preferential concentration of light and heavy particles. Furthermore, we quantitatively show that extreme fluctuations, leading to deviations from a normal-distributed statistics, result from the entangled interaction of the vertical structures with the turbulent background. Finally, we present the first-ever measurement of the relative importance between Stokes drag, Coriolis force and centripetal forces along the trajectories of inertial particles. We discover that vortical coherent structures lead to unexpected diffusion properties for heavy and light particles in the directions parallel and perpendicular to the rotation axis.
In this paper a general theory for interpolation methods on a rectangular grid is introduced. By the use of this theory an efficient B-spline based interpolation method for spectral codes is ...presented. The theory links the order of the interpolation method with its spectral properties. In this way many properties like order of continuity, order of convergence and magnitude of errors can be explained. Furthermore, a fast implementation of the interpolation methods is given. We show that the B-spline based interpolation method has several advantages compared to other methods. First, the order of continuity of the interpolated field is higher than for other methods. Second, only one FFT is needed whereas e.g. Hermite interpolation needs multiple FFTs for computing the derivatives. Third, the interpolation error almost matches the one of Hermite interpolation, a property not reached by other methods investigated.
The hydrodynamic forces exerted by a fluid on small isolated rigid spherical particles are usually well described by the Maxey-Riley (MR) equation. The most time-consuming contribution in the MR ...equation is the Basset history force which is a well-known problem for many-particle simulations in turbulence. In this paper a novel numerical approach is proposed for the computation of the Basset history force based on the use of exponential functions to approximate the tail of the Basset force kernel. Typically, this approach not only decreases the cpu time and memory requirements for the Basset force computation by more than an order of magnitude, but also increases the accuracy by an order of magnitude. The method has a temporal accuracy of O(Delta t^2) which is a substantial improvement compared to methods available in the literature. Furthermore, the method is partially implicit in order to increase stability of the computation. Traditional methods for the calculation of the Basset history force can influence statistical properties of the particles in isotropic turbulence, which is due to the error made by approximating the Basset force and the limited number of particles that can be tracked with classical methods. The new method turns out to provide more reliable statistical data.
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