Analyzing the effect of dilatation on the velocity gradient tensor using a model problem

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.

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 [3,4,5].

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, ${\mathsf {A}} = {\mathsf {\nabla }} {\mathsf {u}}$, is described by the following equation:

$$\begin{aligned} \frac{\hbox {D}A_{ij}}{\hbox {D}t}=-A_{i \alpha } A_{\alpha j} - \Pi _{ij}, \end{aligned}$$

(1)

where the $\Pi _{ij}$’s are the components of the pressure Hessian tensor, ${\mathsf {\Pi }} = {\mathsf {\nabla }}[({\mathsf {\nabla }} p)/\rho ]$, with p and $\rho$ being, respectively, the pressure and the mass density.

In the two-dimensional case, Eq. (1) can be expressed by a four-equation system:

$$\begin{aligned} \frac{\hbox {D} \sigma _n}{\hbox {D} t}= & {} -\delta \sigma _n + \Pi _{22} - \Pi _{11}, \end{aligned}$$

(2)

$$\begin{aligned} \frac{\hbox {D} \sigma _s}{\hbox {D} t}= & {} -\delta \sigma _s - \Pi _{12} - \Pi _{21}, \end{aligned}$$

(3)

$$\begin{aligned} \frac{\hbox {D} \omega }{\hbox {D} t}= & {} -\delta \omega + \Pi _{12} - \Pi _{21}, \end{aligned}$$

(4)

$$\begin{aligned} \frac{\hbox {D}P}{\hbox {D} t}= & {} -\frac{1}{2}(\sigma ^2-\omega ^2+P^2) - \Pi _{11} - \Pi _{22}, \end{aligned}$$

(5)

where $\delta (t)$ is the expansion—or dilatation—rate, $\delta (t) = -1/\rho \centerdot \hbox {D}\rho /\hbox {D}t$, $\sigma _n = A_{11}-A_{22}$ and $\sigma _s = A_{12}+A_{21}$ are, respectively, the normal and shear components of strain, $\sigma = (\sigma _n^2 + \sigma _s^2)^{1/2}$ is the strain intensity, $\omega = A_{21}-A_{12}$ is the vorticity, and $P = A_{11}+A_{22}$ is the velocity divergence which—as a result of mass conservation—coincides with the dilatation rate: $P \equiv \delta$.

The model problem is based on assuming the expansion rate as:

$$\begin{aligned} \delta (c) = 4 \delta _m c(1-c), \end{aligned}$$

(6)

with $c(t) = (\rho _o/\rho (t) - 1)/(\rho _o/\rho _{\infty } - 1)$ where $\rho _o \equiv \rho (0)$, and $\rho _{\infty } \equiv \lim \limits _{t \rightarrow \infty } \rho (t)$; the density ratio is defined by $\rho _o/\rho _{\infty }$.The parabolic function modeling $\delta (c)$ is inspired from numerical simulation data for the velocity divergence across a flame front [7]. From the definitions of c(t) and $\delta (t)$:

$$\begin{aligned} \frac{\hbox {D}c}{\hbox {D}t}=\left( c + \frac{1}{q} \right) \delta , \end{aligned}$$

(7)

with $q = \rho _o/\rho _{\infty } - 1$. In this study, we state $\delta _m = q$, with $q > 0$, which means $\rho _{\infty } < \rho _o$—and $\delta > 0$—as a result, for instance, of heat release. Note that from the approach of Tien and Matalon [8] $\delta \simeq q/\tau _f$ in the reaction zone of a flame front, where $\tau _f$ is the flame timescale. Stating $\delta _m = q$ thus comes to normalize $\delta _m$ by $1/\tau _f$. This choice is convenient, for it makes $\delta (c)$ depend on a single parameter. Figure 1 shows c(t) and $\delta (t)$ for $q = 5$ and $q = 1$.

Fig. 1

Evolution of c and $\delta$; (1) c(t) for $q = 5$; (2) for $q = 1$; (3) $\delta (t)$ for $q = 5$; (4) for $q = 1$

The evolution of the velocity tensor is computed from Eqs. (2)–(4), with the EHEE modeled equation for $\Pi _{12}$, $\Pi _{21}$, and $\Pi _{22}$ [6]:

$$\begin{aligned} \frac{\hbox {D}\Pi _{ij}}{\hbox {D} t}=-A_{\alpha j} \Pi _{i \alpha } - A_{\alpha i} \Pi _{\alpha j}- (n-1) A_{\alpha \alpha } \Pi _{ij}, \end{aligned}$$

(8)

where n is the ratio of specific heats, while component $\Pi _{11}$ is computed from Eq. (5) with P derived from Eqs. (6) and (7).

A study spanning a range of initial conditions is not within the scope of this work. As a first step, the physical relevance of the model is checked with a single set of initial conditions, namely: $c(0) = 10^{-4}$, $\sigma _n(0) = -\delta _m$, $\sigma _s(0) = 0.1 |\sigma _n(0)|$, $\omega (0) = \sigma (0)$, together with isotropy of tensor ${\mathsf {\Pi }}$, namely $\Pi _{12}(0) = \Pi _{21}(0) = 0$, and $\Pi _{11}(0)$ and $\Pi _{22}(0)$ derived from Eq. (5) at $t = 0$, with $\Pi _{11}(0) = \Pi _{22}(0)$.

In Fig. 2, $A_{11}$, $A_{22}$, and $\delta /A_{22}$ are plotted against c, for $q = 5$. Interestingly, the behavior shown in Fig. 2 is akin to the structure of normal strain ($a_N \equiv A_{11}$) and tangential strain ($a_T \equiv A_{22}$) across a flame front [7, 9].

Fig. 2

Diagonal components of ${\mathsf {A}}$, (1) $A_{11}$, (2) $A_{22}$, (3) ratio $\delta /A_{22}$, versus c, for $q = 5$

3 Effect of dilatation on strain structure

The evolution of strain tensor properties, namely strain eigenvalues, $\lambda _1 = (-\sigma + \delta )/2$ and $\lambda _2 = (\sigma + \delta )/2$ [10] as well as orientation of the strain eigenvectors, ${\mathsf {e}}_1$ and ${\mathsf {e}}_2$, is examined for both $q = 5$ (large density ratio) and $q = 1$ (low density ratio ) with the same latter initial conditions.

3.1 Large density ratio

Because it determines the sign of the lowest strain eigenvalue, the ratio of dilatation rate to strain intensity, $\delta /\sigma$, is a significant parameter of the dynamics of a scalar gradient in a non-solenoidal flow [10]. As shown in Fig. 3, dilatation makes the smallest eigenvalue, $\lambda _1$, positive—which thus means two extensional strain directions—over most of the c-range, where $\delta /\sigma > 1$. It is only at the edges ($c < 0.12$, and $c > 0.92$), where $\delta /\sigma < 1$, that $\lambda _1 < 0$, which thus leads to one compressional and one extensional strain directions.

Fig. 3

Ratio $\delta /\sigma$ and sign of strain eigenvalues versus c; $q = 5$

The orientation of strain eigenvectors is shown by $\Phi = \arctan (\sigma _n/\sigma _s)/2 - \pi /4$, the angle between axis ${\mathsf {x}}_1$ and the direction of the largest strain, ${\mathsf {e}}_2$ (Fig. 4). For small c, $\Phi < -\pi /4$, which means that direction ${\mathsf {x}}_1$ mostly undergoes the influence of the smallest strain. As c increases, counterclockwise rotation of strain axes brings the direction of the largest strain near ${\mathsf {x}}_1$, and this orientation is hold all over the intermediate c-range. As c reaches the upper range, rotation of strain axes is reversed, and the direction of the lowest strain comes back close to ${\mathsf {x}}_1$. These changes in strain axes orientation, in particular, alignment of the largest strain with the direction of anisotropy, are consistent with the evolution of strain approaching a flame front [5, 11].

Fig. 4

Angle $\Phi$ between ${\mathsf {x}}_1$ and the direction of largest strain, ${\mathsf {e}}_2$, versus c; $q = 5$; a solid arrow indicates compressional strain, while a dashed arrow indicates extensional strain; direction of the largest strain is shown by a bold dashed arrow

In this two-dimensional Euler flow, rotation of strain principal axes is promoted by anisotropy of the pressure Hessian tensor [12]. The rotation rate of strain eigenvectors is indeed given by $\Omega = 2 \hbox {D}\Phi /\hbox {D}t = \sigma ^{-2}(\sigma _s \hbox {D}\sigma _n/\hbox {D}t-\sigma _n \hbox {D}\sigma _s/\hbox {D}t)$, and then, from Eqs. (2) and (3), $\Omega = \sigma ^{-2}[\sigma _s(\Pi _{22}-\Pi _{11})+\sigma _n(\Pi _{12}+\Pi _{21})]$. Figure 5 clearly shows the anisotropy of ${\mathsf {\Pi }}$ revealed by $\Pi _{11}$ prevailing over the other components, and the resulting rotation rate, $\Omega$.

Fig. 5

Rotation rate of strain principal axes, $\Omega$, and components of pressure Hessian tensor, $\Pi _{ij}$, versus c; $q = 5$; (1) $\Omega$; (2) $\Pi _{11}$; (3) $\Pi _{12}$ and $\Pi _{21}$; (4) $\Pi _{22}$

3.2 Low density ratio

Similar features of strain evolution are retrieved for $q = 1$ with, however, a lesser influence of expansion rate. The lowest strain eigenvalue again gets positive for intermediate c values as expansion rate exceeds strain intensity, but this time over a somewhat shorter range (Fig. 6).

The influence of density ratio is more obvious in strain axes orientation (Fig. 7). The direction of the largest strain comes much less close to ${\mathsf {x}}_1$ over the intermediate c-range, and the latter is more narrow as well. This directly results from a weaker rotation rate, $\Omega$, for $q = 1$ (not shown). Indeed the lower q, the smaller the respective magnitudes of strain components, $\sigma _n$ and $\sigma _s$, and of anisotropic terms, $\Pi _{22} - \Pi _{11}$ and $\Pi _{12} + \Pi _{21}$; and the level of $\sigma ^{-2}$—greater for $q = 1$ than for $q = 5$—is not enough to balance this difference.

Fig. 6

Ratio $\delta /\sigma$ and sign of strain eigenvalues versus c; $q = 1$

Fig. 7

Angle $\Phi$ between ${\mathsf {x}}_1$ and the direction of largest strain, ${\mathsf {e}}_2$, versus c; $q = 1$; a solid arrow indicates compressional strain, while a dashed arrow indicates extensional strain; direction of the largest strain is shown by a bold dashed arrow

4 Conclusion

The mechanisms underlying the influence of dilatation on the velocity gradient tensor can be reliably addressed with a model problem. Assuming a likely evolution of the expansion rate, the solution of an equation system for the components of the velocity gradient and of the pressure Hessian includes a number of features regarding the evolution of the dynamic field as a density front is approached.

The ratio of dilatation rate to strain intensity, a critical parameter in the variable-mass-density kinematics of scalar gradient, is derived. More specifically, the evolution of normal and tangential strains is reminiscent of strain structure at the crossing of a flame front. Finally, the pressure Hessian anisotropy resulting from forcing the expansion rate promotes the rotation of strain principal axes, which subsequently aligns the largest strain with the direction of anisotropy, a result relevant to questions at issue in the physics of flames. Extension of the approach to the three-dimensional case as well as to the coupling of the velocity gradient dynamics with the physics of a reacting scalar gradient is a work in progress.