Problems with an Incompressibility Constraint

The standard practice in the literature for modeling materials processing in which changes in temperature induce significant volume changes is based on the a posteriori substitution of a temperature-dependent expression for density into the governing equations for an incompressible material. In this paper we show this ad hoc approach misses important terms in the equations, and by example show the ad hoc equations fail to capture important physical effects. First we derive the three-dimensional equations which govern the deformation and heat transfer of materials with prescribed temperature-dependent density. Specification of density as a function of temperature translates to a thermomechanical constraint, in contrast to the purely mechanical incompressibility constraint, so that the constraint response function (“pressure”) enters into the energy equation as well as the momentum equation. Then we demonstrate the effect of the correct constraint response by comparing solutions of our thermomechanical theory with solutions of the ad hoc theory in plane Poiseuille flow. The differences are significant, both quantitatively and qualitatively. In particular, the observed phenomenon of expansion cooling is captured by the thermomechanically constrained theory, but not by the ad hoc theory.

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Wave-vortex dynamics in rotating shallow water

Journal of Fluid Mechanics ◽

10.1017/s0022112089002351 ◽

1989 ◽

Vol 206 ◽

pp. 433-462 ◽

Cited By ~ 66

Author(s):

Marie Farge ◽

Robert Sadourny

Keyword(s):

Shallow Water ◽

Turbulent Flows ◽

Gravitational Energy ◽

Energy Cascade ◽

Adjustment Process ◽

Coherent Vortices ◽

Small Scales ◽

Direct Energy ◽

Incompressibility Constraint ◽

Rotating Shallow Water

We investigate how two-dimensional turbulence is modified when the incompressibility constraint is removed, by numerically integrating the full Saint-Venant (shallow-water) equations. In the case of small geopotential fluctuations considered here, we find no energy exchange between the inertio-gravitational and the potentio-vortical components of the flow. At small scales, the potentio-vortical component behaves as if the flow were incompressible, while we observe an intense direct energy cascade within the inertio-gravitational component. At large scales, the reverse potentio-vortical energy cascade is reduced when the level of inertio-gravitational energy is high. Looking at the effect of rotation, we find that a fast rotation rate tends to inhibit all three cascades. In particular, the inhibition of the inertio-gravitational energy cascade towards small scales implies that the geostrophic adjustment process is hindered by an increase of rotation. Concerning the structure of the coherent vortices emerging out of these decaying turbulent flows, we observe that the smallest scales are concentrated inside the vortex cores and not on their periphery.

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A note on relaxation with constraints on the determinant

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2018030 ◽

2019 ◽

Vol 25 ◽

pp. 41 ◽

Cited By ~ 1

Author(s):

Marco Cicalese ◽

Nicola Fusco

Keyword(s):

Calculus Of Variations ◽

Upper Bound ◽

Lower Semicontinuity ◽

Carathéodory Function ◽

Incompressibility Constraint ◽

Multiple Integrals ◽

Quasiconvex Envelope

We consider multiple integrals of the Calculus of Variations of the form E(u) = ∫ W(x, u(x), Du(x)) dx where W is a Carathéodory function finite on matrices satisfying an orientation preserving or an incompressibility constraint of the type, det Du > 0 or det Du = 1, respectively. Under suitable growth and lower semicontinuity assumptions in the u variable we prove that the functional ∫ Wqc(x, u(x), Du(x)) dx is an upper bound for the relaxation of E and coincides with the relaxation if the quasiconvex envelope Wqc of W is polyconvex and satisfies p growth from below for p bigger then the ambient dimension. Our result generalises a previous one by Conti and Dolzmann [Arch. Rational Mech. Anal. 217 (2015) 413–437] relative to the case where W depends only on the gradient variable.

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Fluid-Structure Interaction Simulation With Free Surface Flows by Smoothed Particle Hydrodynamics

Volume 12: Mechanics of Solids, Structures and Fluids ◽

10.1115/imece2008-66769 ◽

2008 ◽

Cited By ~ 1

Author(s):

M. H. Farahani ◽

N. Amanifard ◽

H. Asadi ◽

M. Mahnama

Keyword(s):

Free Surface ◽

Fluid Structure Interaction ◽

Free Surface Flows ◽

Surface Flows ◽

Fluid Structure ◽

Structure Interaction ◽

Particle Hydrodynamics ◽

Smoothed Particle ◽

Incompressibility Constraint ◽

Sph Algorithm

Simulation of the fluid-structure interaction (FSI) and free surface flows includes an area of extremely challenging problems in the computational mechanics community. In this paper, a newly proposed SPH algorithm is employed to simulate FSI problems with complex free surface flows. In this way, fluid and elastic structure continua are coupled using a monolithic but explicit numerical scheme. The proposed method is similar to so-called SPH projection method and consists of three steps. The first two steps play the role of prediction, while in the third step a Poisson equation is used for both fluid and structure to impose incompressibility constraint.

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A TV-Based Iterative Regularization Method for the Solutions of Thermal Convection Problems

Communications in Computational Physics ◽

10.4208/cicp.270212.310113a ◽

2013 ◽

Vol 14 (4) ◽

pp. 1120-1146

Author(s):

Eric T. Chung ◽

Jeff C.-F. Wong

Keyword(s):

Stokes Problem ◽

Fluid Flows ◽

Iterative Regularization ◽

Image Diffusion ◽

Incompressibility Constraint ◽

Spurious Oscillations ◽

Convective Fluid ◽

Artificial Pressure ◽

Tv Model ◽

Selection Of

AbstractLinear/nonlinear and Stokes based-stabilizations for the filter equations for damping out primitive variable (PV) solutions corrupted by uniformly distributed random noises are numerically studied through the natural convection (NC) as well as the mixed convection (MC) environment. The most recognizable filter-scheme is based on a combination of the negative Laplace equation multiplied with the selection of the spatial scale and a linear function in order to preserve the uniqueness of the filtered solution. A more complicated filter-scheme, based on a Stokes problem which couples a filtered velocity and a filtered (artificial) pressure (or Lagrange multiplier) in order to enforce the incompressibility constraint, is also studied. Linear and Stokes based-filters via nested iterative (NI) filters and the consistent splitting scheme (CSS) are proposed for the NC/MC problems. Inspired by the total-variation (TV) model of image diffusion, well preserved feature flow patterns from the corrupted NC/MC environment are obtained by TV-Stokes based-filters together with the CSS. Our experimental results show that our proposed algorithms are effective and efficient in eliminating the unwanted spurious oscillations and preserving the accuracy of thermal convective fluid flows.

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