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This paper extends the derivation of the Lagrangian averaged Euler (LAE-$alpha$) equations to the case of barotropic compressible flows. The aim of Lagrangian averaging is to regularize the compressible Euler equations by adding dispersion instead of artificial viscosity. Along the way, the derivation of the isotropic and anisotropic LAE-$alpha$ equations is simplified and clarified. The derivation in this paper involves averaging over a tube of trajectories $eta^epsilon$ centered around a given Lagrangian flow $eta$. With this tube framework, the Lagrangian averaged Euler (LAE-$alpha$) equations are derived by following a simple procedure: start with a given action, Taylor expand in terms of small-scale fluid fluctuations $xi$, truncate, average, and then model those terms that are nonlinear functions of $xi$. Closure of the equations is provided through the use of emph{flow rules}, which prescribe the evolution of the fluctuations along the mean flow.
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The need to develop models to predict the motion of microrobots, or robots of a much smaller scale, moving in fluids in a low Reynolds number regime, and in particular, in non Newtonian fluids, cannot be understated. The article develops a Lagrangian based model for one such mechanism - a two-link mechanism termed a microscallop, moving in a low Reynolds number environment in a non Newtonian fluid. The modelling proceeds through the conventional Lagrangian construction for a two-link mechanism and then goes on to model the external fluid forces using empirically based models for viscosity to complete the dynamic model. The derived model is then simulated for different initial conditions and key parameters of the non Newtonian fluid, and the results are corroborated with a few existing experimental results on a similar mechanism under identical conditions. Lastly, with a view to implementing control algorithms we explore accessibility of the system at certain configurations.
The interplay between incompressibility and stratification can lead to non-conservation of horizontal momentum in the dynamics of a stably stratified incompressible Euler fluid filling an infinite horizontal channel between rigid upper and lower plates. Lack of conservation occurs even though in this configuration only vertical external forces act on the system. This apparent paradox was seemingly first noticed by Benjamin (J. Fluid Mech., vol. 165, 1986, pp. 445-474) in his classification of the invariants by symmetry groups with the Hamiltonian structure of the Euler equations in two dimensional settings, but it appears to have been largely ignored since. By working directly with the motion equations, the paradox is shown here to be a consequence of the rigid lid constraint coupling through incompressibility with the infinite inertia of the far ends of the channel, assumed to be at rest in hydrostatic equilibrium. Accordingly, when inertia is removed by eliminating the stratification, or, remarkably, by using the Boussinesq approximation of uniform density for the inertia terms, horizontal momentum conservation is recovered. This interplay between constraints,action at a distance by incompressibility, and inertia is illustrated by layer-averaged exact results, two-layer long-wave models, and direct numerical simulations of the incompressible Euler equations with smooth stratification.
Embedding geometries in structured grids allows a simple treatment of complex objects in fluid flows. Various methods are available. The commonly used Brinkman-volume-penalization models geometries as porous media, where in the limit of vanishing porosity a solid object is approximated. In the simplest form, the velocity equations are augmented by a term penalizing the fluid velocity, the body velocity. yielding good results in many applications. However, it induces numerical stiffness, especially if high pressure gradients need to be balanced. Here, we focus on the effect of the reduced effective volume (commonly called porosity) of the porous medium. An approach is derived, which allows to reduce the flux through objects to practically zero with little increase of numerical stiffness. Also, non-slip boundary conditions and adiabatic boundary conditions are easily constructed. The porosity terms allow to keep the skew symmetry of the underlying numerical scheme, by which the numerical stability is improved. Furthermore, a very good conservation of mass and energy in the non-penalized domain can be achieved. The scheme is tested for acoustic scenarios, near incompressible and strongly compressible flows.
Common modal decomposition techniques for flowfield analysis, data-driven modeling and flow control, such as proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are usually performed in an Eulerian (fixed) frame of reference with snapshots from measurements or evolution equations. The Eulerian description poses some difficulties, however, when the domain or the mesh deforms with time as, for example, in fluid-structure interactions. For such cases, we first formulate a Lagrangian modal analysis (LMA) ansatz by a posteriori transforming the Eulerian flow fields into Lagrangian flow maps through an orientation and measure-preserving domain diffeomorphism. The development is then verified for Lagrangian variants of POD and DMD using direct numerical simulations (DNS) of two canonical flow configurations at Mach 0.5, the lid-driven cavity and flow past a cylinder, representing internal and external flows, respectively, at pre- and post-bifurcation Reynolds numbers. The LMA is demonstrated for several situations encompassing unsteady flow without and with boundary and mesh deformation as well as non-uniform base flows that are steady in Eulerian but not in Lagrangian frames. We show that LMA application to steady nonuniform base flow yields insights into flow stability and post-bifurcation dynamics. LMA naturally leads to Lagrangian coherent flow structures and connections with finite-time Lyapunov exponents (FTLE). We examine the mathematical link between FTLE and LMA by considering a double-gyre flow pattern. Dynamically important flow features in the Lagrangian sense are recovered by performing LMA with forward and backward (adjoint) time procedures.
We study numerically joint mixing of salt and colloids by a chaotic velocity field $mathbf{V}$, and how salt inhomogeneities accelerate or delay colloid mixing by inducing a velocity drift $mathbf{V}_{rm dp}$ between colloids and fluid particles as proposed in recent experiments cite{Deseigne2013}. We demonstrate that because the drift velocity is no longer divergence free, small variations to the total velocity field drastically affect the evolution of colloid variance $sigma^2=langle C^2 rangle - langle C rangle^2$. A consequence is that mixing strongly depends on the mutual coherence between colloid and salt concentration fields, the short time evolution of scalar variance being governed by a new variance production term $P=- langle C^2 abla cdot mathbf{V}_{rm dp} rangle/2$ when scalar gradients are not developed yet so that dissipation is weak. Depending on initial conditions, mixing is then delayed or enhanced, and it is possible to find examples for which the two regimes (fast mixing followed by slow mixing) are observed consecutively when the variance source term reverses its sign. This is indeed the case for localized patches modeled as gaussian concentration profiles.
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