Tensor algebra

More advanced models, for example the algebraic stress model (ASM) and the Reynolds stress model (RSM), are not based on the eddy-viscosity concept and can thus account for anisotropic turbulence thereby giving still better predictions of flows. In addition to the transport equations, however, the algebraic equations for the Reynolds stress tensor also have to be solved. These models are therefore computationally far more complex than simple closure models (Kuipers and van Swaaij, 1997). 

Matrix and tensor notation is useful when dealing with systems of equations. Matrix theory is a straightforward set of operations for linear algebra and is covered in Section A.I. Tensor notation, treated in Section A.2, is a classification scheme in which the complexity ranges upward from scalars (zero-order tensors) and vectors (first-order tensors) through second-order tensors and beyond. 

WilS73a Williamson, S. G. Tensor compositions and lists of combinatorial structures. Linear and Multilinear Algebra 1... 

Calculation of the angular part of the matrix elements thus remains, which can be performed exactly using tensor algebra techniques based on group theory. Since the calculation of the matrix elements is not straightforward, we provide here some details on it for the interested reader. The treatment follows the procedure described in Ref. [17]. 

This type of model is usually referred to as an algebraic scalar-flux model. Similarmodels for the Reynolds-stress tensor are referred to as algebraic second-moment (ASM) closures. They can be derived from the scalar-flux transport equation by ignoring time-dependent and spatial-transport terms. 

The next important problem in algebraic theory is the construction of the basis states (the representations) on which the operators X act. A particular role is played by the irreducible representations (Appendix A), which can be labeled by a set of quantum numbers. For each algebra one knows precisely how many quantum numbers there are, and a list is given in Appendix A. The quantum numbers are conveniently arranged in patterns (or tableaux), called Young tableaux. Tensor representations of Lie algebras are characterized by a set of integers... 

The algebra of U(4) can be written in terms of spherical tensors as in Table 2.1. This is called the Racah form. The square brackets in the table denote tensor products, defined in Eq. (1.25). Note that each tensor operator of multipolarity X has 2X+ 1 components, and thus the total number of elements of the algebra is 16, as in the uncoupled form. 

On account of the hermiticity of the operators, only even values of L are allowed. Since the D operators are tensor operators of rank 1, the only allowed values are L = 0,2. The L = 0 contribution has been treated in type (1). Hence, only the L = 2 contribution must be considered here. The matrix elements of the operators (4.122) with L = 2 are difficult to evaluate. Nonetheless, by making use of the angular momentum algebra, they can be evaluated in explicit... 

Table A.4 Number of integers that characterize the tensor representations of Lie algebras...

We return to the simple example of the angular momentum algebra, SO(3). Its tensor representations are characterized by one integer (Table A.4), that is, the angular momentum quantum number J. Similarly, the representations of SO(2) are characterized by one integer (Table A.4) that is, M the projection of the angular momentum on the z axis. The complete chain of algebras is... 

A tensor operator under the algebra G 3 G, T, is defined as that operator satisfying the commutation relations... 

The celebrated Wigner-Eckart theorem states that the matrix elements of any tensor operator of an algebra G can be split into two pieces, a coupling coefficient and a piece that depends only on A that is,... 

We then compare Eq. (2.418) to the second line on the RHS of Eq. (2.390), for the case of a generalized projection tensor = P , which is the same as the inertial or geometric projection tensor in this simple class of models. After some straightforward algebra, we find that, for the class of models to which Liu s algorithm applies, the last term in Eq. (2.390) may be written more explicitly as... 

The use of literate programming methods leads naturally to structure and standardization in computer code. In turn, this structure leads to subroutine libraries and we describe the specification of a basic tensor algebra subroutine library, which we have recently developed, and which we expect to prove useful in a range of applications. 

We propose [15] a set of basis tensor algebra subroutines or btas. Tensors and tensor operators arise in many fields in the computational sciences, including computational quantum chemistry. The nomenclature BTAs(m,n), with m > n, where m and n are the respective ranks of the tensors, is proposed to establish a high level classification of tensor operations. The BTAS can be classified as follows -BTAS(1,0) BTAS(1,1)... 

Let us briefly recall a few of the basics of the algebra of tensors. An nth rank tensor in m-dimensional space is an object with n indices and rrf components. For a general tensor a distinction is made between contravari-ant (upper) indices and covariant (lower) indices. A tensor of rank mi + m2 may have mi contravariant indices and m2 covariant indices. The order of the indices is significant. Tensors can be classified according to whether they are... 

The determination of the order tensor corresponds to the solution of a linear algebraic problem of standard Ax = b form. To see this, consider that Eq. (23) may be expressed in the following form,... 

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