Advances in Mathematical Physics
Volume 2015 (2015), Article ID 147125, 7 pages
http://dx.doi.org/10.1155/2015/147125

Research Article

Simultaneous Invariants of Strain and Rotation Rate Tensors and Their Admitted Region

Igor Vigdorovich¹ and Holger Foysi²

¹Institute of Mechanics, Lomonosov Moscow State University, Michurinsky Avenue 1, Moscow 119192, Russia
²Chair of Fluid Mechanics, Universität Siegen, Paul-Bonatz-Straße 9–11, 57068 Siegen, Germany

Received 11 August 2015; Accepted 29 September 2015

Academic Editor: Christian Engstrom

Copyright © 2015 Igor Vigdorovich and Holger Foysi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The purpose of this paper is to establish the admitted region for five simultaneous, functionally independent invariants of the strain rate tensor and rotation rate tensor and calculate some simultaneous invariants of these tensors which are encountered in the theory of constitutive relations for turbulent flows. Such a problem, as far as we know, has not yet been considered, though it is obviously an integral part of any problem in which scalar functions of the tensors and are studied. The theory provided inside this paper is the building block for a derivation of new algebraic constitutive relations for three-dimensional turbulent flows in the form of expansions of the Reynolds-stress tensor in a tensorial basis formed by the tensors and , in which the scalar coefficients depend on simultaneous invariants of these tensors.

1. Introduction

The strain rate tensor and the rotation rate tensors are the symmetric and antisymmetric parts of the velocity gradient tensor, respectively. In what follows, we will consider incompressible flow, when the tensor is a deviator by virtue of the continuity equation that satisfies the extra condition .

After transforming to principal axes, the tensor has the form

The tensor , in an arbitrary coordinate system, is written as

where , , and are the vorticity vector components. If tensor (2) is written in the coordinate system connected with the principal axes of the tensor , the five quantities

constitute a full set of simultaneous, functionally independent invariants of the tensors and [1].

If a scalar physical quantity depends on the tensors and , it is a function of quantities (3). For example, the theory of algebraic constitutive relations (algebraic stress models) for turbulent flows [2–5] deals with functional relations between the Reynolds-stress tensor and strain and rotation rate tensors that are calculated from the mean velocity field. Such relations are equivalent to the representations of the stress tensor in terms of expansions in a tensorial basis formed by the tensors and . The scalar coefficients in the expansions depend on simultaneous invariants of these tensors.

The simultaneous invariants are conveniently chosen to be quantities that, unlike (3), are easily calculable in an arbitrary coordinate system. For example, they can be the following five quantities that are calculated as the trace of some products of the tensors and :

It is just invariants (4) and their algebraic combinations that are used in the algebraic stress models for turbulent flows (see, e.g., [2–11]). As opposed to invariants (3), which can take arbitrary, independent values, quantities (4) vary in a certain region.

The aim of the present paper is to determine the admitted region for invariants (4). Such a problem, as far as we know, has not yet been considered, though it is obviously an integral part of any problem in which scalar functions of the tensors and are studied. We will calculate in passing some invariants of the tensors and that are encountered in the theory of constitutive relations for turbulent flows as functions of invariants (4). In the present study, the tensor is an arbitrary symmetric, traceless tensor and is an arbitrary antisymmetric tensor.

In Section 2, we determine the admitted region for the invariants and of the strain rate tensor and calculate in terms of and the invariant , which is needed in what follows. In Section 3, we calculate the important simultaneous invariant of the tensors and   , which in a sense is independent of invariants (4), since it can be determined in terms of them only with ambiguity in sign. In Section 4, on the basis of the representation for the invariant , the admitted region for invariants (4) is determined.

To calculate new invariants in terms of quantities (4), two basic relationships are used: the Cayley-Hamilton identity and finite isotropic relations between two tensors, which exist, if the tensors have a diagonal form in one coordinate system [1].

2. Invariants of the Tensor

For the traceless tensor , the Cayley-Hamilton identity [1] reads

where is the identity matrix. After calculating the trace of both sides of equality (5), we get

To this equality, we add two more

The latter is obtained after squaring equality (7) and making use of the definition of the invariant . Equations (6)–(8) imply that the eigenvalues owing to the Viete theorem are the roots of the cubic equation

and therefore have real values under the condition

which specify the admitted region for two invariants of the strain rate tensor with zero trace.

To calculate the invariant

we consider the matrix

Since and (1) have a diagonal form in the same coordinate system, they should be related by an isotropic relation [1]

where , , and are scalar coefficients. By calculating the trace of both sides of equality (13) and then multiplying it consecutively by and and taking the trace, we obtain the system of equations

Here, we used the equalities

and , the latter of which is obtained by multiplying the Cayley-Hamilton identity (5) by and taking the trace.

The calculation of the inverse matrix gives

which enables us to determine the coefficients , , and and rewrite equality (13) in the form

Multiplying (17) by , taking the trace, and using again equalities (15) as well as the equality , which follows from (8), we obtain the representation in question:

This formula also implies condition (10).

3. Invariant

In what follows, we will need the invariants

which are formed by analogy to invariants (4). Multiplying the Cayley-Hamilton identity (5) consecutively by and , taking the trace, and using equalities (4), we get

Now the representation of invariants (19) in terms of invariants (4) can be written in the form of the matrix equality

Consider the invariant

the direct calculation of which gives .

The matrixes

and should be also related by the isotropic relation

Calculating the trace of both sides of equality (24) and then multiplying this equality consecutively by and , and taking the trace, we obtain the system of equations

with the same matrix (14) on the left-hand side. The coefficients in question are specified by the equality

where the matrix has the form (16).

Multiplying equality (24) by the matrix

and calculating the trace yield

Since the coefficients are specified by equality (26), the invariant can be determined in terms of the invariants .

Direct calculations give

On the other hand, the left-hand sides of equalities (30) can be obtained on the basis of the Cayley-Hamilton identity for the tensor , which reads . By multiplying it consecutively by , , and , taking the trace, and making use of (4), we get

Equations (30) and (32) imply

Here, we take into account that the first equation of system (25) reads .

Now, let us calculate the quantities specified by (31). Multiplying (24) by , , calculating the trace, and taking into account equalities (4) and (19) and the fact that , we have

Hence, in particular, we get the equality , which reduces (33) to the form

The quantity is the quadratic form of the matrix :

Equalities (21) and (34) imply

where all the indices take the values 0, 1, and 2. Hence, taking the values of the entries of the matrix (21) into account, we get

By substituting into (35) equalities (36) and (38) and the values of that follow from (16) and (26), we finally obtain

It is easy to make sure that (39) coincides with the expression that was given in [12] without deduction. As is seen in (39), the invariant can be determined in terms of invariants (4) but with ambiguity in sign.

4. Admitted Region for the Invariants

Equality (39) enables us to establish the admitted region for invariants (4). Polynomial (39) is homogenies in the variables , , and . By equating it to zero, we obtain an equation which specifies a conic surface in the space with the vertex at the origin. In the variables

(39) takes the form

Here we used equality (18). The parameter according to condition (10) varies over interval .

In accordance with (4), the vectors specified in (21) and are related by the one-to-one relation

This follows from the fact that, in the general case, the determinant is unequal to zero (it vanishes in the only case ). The invariant and therefore polynomial (41) vanish, if at least one component of the vector vanishes. In that case, equality (42) is a parametric representation of a plane in the space . In other words, if , then three quantities , , and should be related by a linear equation of the form

where and are constant coefficients, which specifies a straight line in the plane and a plane in the space . After substituting (43) into (41), we obtain a third-order polynomial in the variable . The polynomial’s coefficients should be equated to zero, which gives the four equations for quantities and :

Owing to the Viete trigonometric formula, the cubic equation (44) has three real roots:

The comparison of (44) with (9) leads to

It is easy to see that if is a root of (44), the other three equations (45) are satisfied at . Thus, polynomial (41) can be factorized as follows:

The straight lines (43) have three points of intersection in the plane with the coordinates

As was said above, on the straight lines (43) in the general case, one component of the vorticity vector vanishes. In points (49), two components are equal to zero. One can make sure of it by turning again to equality (42), which, when the vector has only one nonzero component, is a parametric equation of a straight line that issues out of the origin in the space . Such a line corresponds to a point in the plane . Hence, when the vorticity vector is directed along one of the strain-rate-tensor principal axes, the invariants on the basis of (40) and (49) must be related by the equalities

in which is a root of (44). These relations again imply equalities (47).

After returning to variables (40) and taking into account equalities (47) and the definitions of invariants (4), we rewrite the first factor on the right-hand side of (48) in the form

Here, we used the properties of the roots of (44), which follows from the Viete theorem, particularly, the equality . We get similarly

Now, taking the range of variation of roots (46) into account,

we may deduce that the first and third factors on the right-hand side of (48) are always nonnegative while the second one is less than or equal to zero, and the admitted region for the invariants , , and in the plane is a triangle with apexes in the points (49). In Figure 1(a), these triangle regions are depicted for a range of over the interval . Since polynomial (41) is invariant under simultaneous change in sign of the quantities and , it is sufficient to consider the case . The regions that correspond to positive and negative values of are symmetric with respect to the axis . In the limiting case , the triangle degenerates into a straight-line segment.

(a)

(b)

Figure 1: Admitted regions for the invariants , , and in the planes (a) and (b) for various values of .

Consider the limiting case in detail. Let us introduce new variables

according to which the triangle’s two sides shown in Figure 1(a) become coordinate lines of different families. By solving these equations for and , we obtain

After substituting (55) into equality (48), the latter reads

Now, taking into account the asymptotic behavior of the roots

we can construct the admitted regions for the invariants in the plane , which have the form of right triangles, shown in Figure 1(b).

5. Conclusions

The problem of determining a functional relation between any scalar physical quantity and the strain rate and rotation rate tensors is reduced to determining a function of simultaneous scalar invariants of the tensors and , which from reasons of the ease of computation can be chosen as the quantities , , , , and . Hence, the problem of investigating the admitted region for these five invariants naturally arises. It has been established that this region can be constructed in three-dimensional space of the parameters

where it is situated between the planes , while its boundary in each cross section is a triangle in the plane . When the parameter tends to its limiting values , the triangle’s area vanishes while the triangle degenerates into a straight-line segment.

In the sequel, we plan to derive new algebraic constitutive relations (explicit algebraic stress models) for three-dimensional turbulent flows in the form of expansions of the Reynolds-stress tensor in a tensorial basis formed by the tensors and , in which the scalar coefficients depend on simultaneous invariants of these tensors. The result obtained in the present study will be used, in particular, to investigate the asymptotic behavior of the scalar coefficients at the boundary of the admitted region for the invariants.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was partially supported by the visiting grant of the Deutsche Forschungsgemeinschaft (German Research Foundation) Project no. DFG FO 674/7-1.

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