Analyzing the effect of dilatation on the velocity gradient tensor using a model problem
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Analyzing the effect of dilatation on the velocity gradient tensor using a model problem M. Gonzalez1 Received: 31 March 2020 / Accepted: 15 September 2020 © Springer Nature Switzerland AG 2020
Abstract The effect of variable mass density on the velocity gradient tensor is addressed by means of a model problem. An equation system for both the velocity gradient and the pressure Hessian tensor is solved assuming a realistic expansion rate. The model results show the evolution of the velocity gradient tensor as the density front is approached and are relevant to the physics of flame fronts. Keywords Variable mass density · Velocity gradient tensor · Pressure Hessian · Strain properties · Flame fronts
1 Introduction The influence of local mass density variations upon the properties of the velocity gradient tensor is especially significant in compressible flows or in reacting flows with heat release. Intensity and orientation of both strain and vorticity may be altered, which eventually plays on the growth rate and alignment of scalar gradients. Through the velocity gradient, mass density gradients may thus influence the mixing process, a phenomenon addressed in compressible turbulence [1, 2] and in turbulent flames [3–5]. Such indirect effects often stem from an intricate interaction. For instance, there is now some evidence that, to a large extent, the small-scale physics of turbulent flames is governed by the interplay of the respective gradients of velocity, concentration, and mass density. Explaining the resulting phenomena may thus require, as a first step, analyzing each underlying mechanism separately. The present work is based on this kind of approach. The basic model problem is the evolution of the velocity gradient tensor undergoing a given expansion rate. This is a one-way coupling in which heat release, for instance, is
forced in a restricted flow region, and subsequently affects the velocity gradient properties. The equation system for the velocity gradient tensor, including the enhanced homogenized Euler equation (EHEE) model of Suman and Girimaji [6] for the pressure Hessian tensor, is solved in a two-dimensional Euler flow (Sect. 2). The evolution of strain structure is analyzed for large and low values of the density ratio (Sect. 3).
2 Model problem In an Euler flow, the evolution of the velocity gradient tensor, 𝖠 = ∇𝗎 , is described by the following equation:
DAij Dt
(1)
= −Ai𝛼 A𝛼j − Πij ,
where the Πij ’s are the components of the pressure Hessian tensor, = �[(�p)∕𝜌] , with p and 𝜌 being, respectively, the pressure and the mass density. In the two-dimensional case, Eq. (1) can be expressed by a four-equation system:
* M. Gonzalez, [email protected] | 1CNRS, UMR 6614 CORIA, Site universitaire du Madrillet, 76801 Saint‑Etienne du Rouvray, France. SN Applied Sciences
(2020) 2:1793
| https://doi.org/10.1007/s42452-020-03513-4
Vol.:(0123456789)
Research Article
SN Applied Sciences
(2020) 2:1793
D𝜎n = − 𝛿𝜎n + Π22 − Π11 , Dt
(2)
D𝜎s = − 𝛿𝜎s − Π12 − Π21 , Dt
(3)
D𝜔 = − 𝛿𝜔
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