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Concept# Incompressible flow

Summary

In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An equivalent statement that implies incompressibility is that the divergence of the flow velocity is zero (see the derivation below, which illustrates why these conditions are equivalent).
Incompressible flow does not imply that the fluid itself is incompressible. It is shown in the derivation below that (under the right conditions) even compressible fluids can – to a good approximation – be modelled as an incompressible flow.
Derivation
The fundamental requirement for incompressible flow is that the density, \rho , is constant within a small element volume, dV, which moves at the flow velocity u. Mathematically, this constraint implies that the material derivative (discussed below) of the density must vanish to ensure inc

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In a region D in R-2 or R-3, the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by partial derivative(t)v + (v . del(x))v = -del(xP), div(x)v = 0, where v(t, x) is the velocity of the particle located at x is an element of D at time t and p(t, x) is an element of R is the pressure. Solutions v and p to the Euler equation can be obtained by solving {del x {partial derivative t phi(t, x, a) + p(t, x) + (1/2)|del(x)phi(t, x, a)|(2)} = 0 at a = kappa(t, x), v(t, x) = del(x)phi(t, x, a) at a = kappa(t, x), partial derivative(t)kappa(t, x) | (v. del(x))kappa(t, x) = 0 div(x)v(t, x) = 0 where phi : R x D x R-l -> R and kappa : R x D -> R-l are additional unknown mappings (l >= 1 is prescribed). The third equation in the system says that kappa is an element of R-l is convected by the flow and the second one that phi can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition kappa(0, x) = x on D (and thus l = 2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52: 411-452, 1999) in his Eulerian-Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross. D and that carry each "particle" at time t = 0 at a prescribed location at time t = T > 0, that is, kappa(T, x) is prescribed in D for all x is an element of D. We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary partial derivative D of the bounded region D (a two-or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through partial derivative D of particles labelled by each value of. at each point of partial derivative D. One of the main novelties is the introduction of a prescribed "generalized" Bernoulli's function H : R-l -> R, namely, we add to (0.1) the requirement that partial derivative(t)phi(t, x, a) + p(t, x) + (1/2)|del(x)phi(t, x, a)|(2) = H(a) at a = kappa(t, x) with phi, p, kappa periodic in time of prescribed period T > 0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of "Lamb's surfaces" and "isotropic manifolds" in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier's formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional (kappa, v) -> integral(T)(0)integral(D) {1/2|v(t, x)|(2) + H(kappa(t, x))} dtdx defined for kappa and v that are T-periodic in t, such that partial derivative(t)kappa(t, x) + (v . del(x))kappa(t, x) = 0, div(x)v(t, x) = 0, and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize integral(]0, L[x]0,1[) {(1/2)|del psi|(2) + H(psi)}dx for psi is an element of W-1,W-2 (]0, L[x]0,1[) under appropriate boundary conditions, where. is the stream function. For a minimizer, corresponding functions phi and kappa are given in terms of the stream function psi.

Marc Anthony David Habisreutinger

In fluid mechanics, turbulence can occur in very simple flow geometries, for Newtonian fluids and without the need for additional flow conditions such as temperature gradients or chemical reactions. In standard cases, intuitive assumptions on the physics of the subgrid scales coupled with the classical theories of turbulence can be well suited for subgrid modelling in large eddy simulation. However, considering more complex situations such as elastic or plasmas turbulence, the behaviour of the subgrid scales is not clearly identified, certainly not as intuitive and the corresponding theories are not available yet. The question is how to proceed when the functional modelling, which imposes a known behaviour to the subgrid scales of the flow, is not possible. For instance, this issue could be overcome using deconvolution-based subgrid models which aim at a partial recovery of the original quantities from their filtered counterpart. In principle, functional modelling is avoided by attempting to invert the filtering operator applied to the governing equations. However, this apparent advantage is completely lost since these models are usually coupled with auxiliary approaches, directly based on functional modelling, in order to account for the interactions with the scales which are not representable on the coarse spatial discretization used for large eddy simulation. The driving motivation of this work is to suppress the need for this secondary modelling which would allow to extend the use of deconvolution-based models to the large eddy simulation of flows whose behaviour of subgrid scales is not identified. Considering the effects of the coarse numerical discretization as the only effective filter applied to the macroscopic equations, an interpretation of the deconvolution models as a way to approximate the effect of the scales lost by numerical discretization on the resolved scales of the flow is demonstrated. Consequently, a new category of subgrid models, the grid filter models, is defined and gives a theoretical justification to the use of deconvolution models for the entire subgrid modelling process. In this perspective, a general method for the computation of the convolution filter which models the effect of the grid filter on the computable scales of the solution is proposed, thereby addressing the key issue of the numerical discretization in large eddy simulation. This modelling approach is validated performing the large eddy simulation of the incompressible flow of a Newtonian fluid in a lid-driven cubical cavity. Comparisons with classical subgrid models allow to assess the validity of this modelling approach and the suppression of the need for functional modelling. In order to extend the validity domain of the grid filter models, the large eddy simulation of an elastic turbulence problem is envisaged. Numerical simulations of elastic turbulence are limited by numerical instabilities which are particularly stringent at high elasticity. Moreover, the computational burden resulting from the required space-time resolutions is significantly increased as compared to the Newtonian case. Consequently, available direct numerical simulations are restricted to periodic and two-dimensional cases. Among these studies, the large eddy simulation of the viscoelastic Kolmogorov flow is chosen as benchmark problem.

A space-time adaptive algorithm to solve the motion of a rigid disk in an incompressible Newtonian fluid is presented, which allows collision or quasi-collision processes to be computed with high accuracy. In particular, we recover the theoretical result proven in [M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1345-1371], that the disk will never touch the boundary of the domain in finite time. Anisotropic, continuous piecewise linear finite elements are used for the space discretization, the Euler scheme for the time discretization. The adaptive criteria are based on a posteriori error estimates for simpler problems.