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 formThe tensor , in an arbitrary coordinate system, is written aswhere , , 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 quantitiesconstitute 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] readswhere is the identity matrix. After calculating the trace of both sides of equality (5), we getTo this equality, we add two moreThe 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 equationand therefore have real values under the conditionwhich specify the admitted region for two invariants of the strain rate tensor with zero trace.
To calculate the invariantwe 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 equationsHere, we used the equalitiesand , the latter of which is obtained by multiplying the Cayley-Hamilton identity (5) by and taking the trace.
The calculation of the inverse matrix giveswhich enables us to determine the coefficients , , and and rewrite equality (13) in the formMultiplying (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 invariantswhich are formed by analogy to invariants (4). Multiplying the Cayley-Hamilton identity (5) consecutively by and , taking the trace, and using equalities (4), we getNow the representation of invariants (19) in terms of invariants (4) can be written in the form of the matrix equality
Consider the invariantthe direct calculation of which gives .
The matrixes and should be also related by the isotropic relationCalculating 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 equationswith the same matrix (14) on the left-hand side. The coefficients in question are specified by the equalitywhere the matrix has the form (16).
Multiplying equality (24) by the matrixand calculating the trace yield Since the coefficients are specified by equality (26), the invariant can be determined in terms of the invariants .
Direct calculations giveOn 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 getEquations (30) and (32) implyHere, 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 haveHence, in particular, we get the equality , which reduces (33) to the formThe 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 getBy substituting into (35) equalities (36) and (38) and the values of that follow from (16) and (26), we finally obtainIt 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 formHere 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 relationThis 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 formwhere 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 toIt 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 coordinatesAs 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 equalitiesin 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 formHere, we used the properties of the roots of (44), which follows from the Viete theorem, particularly, the equality . We get similarlyNow, 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)
Consider the limiting case in detail. Let us introduce new variablesaccording 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 obtainAfter 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.