Added scalar adjoint advection terms in AdjointSolver

Issue/feature addressed

The scalar (i.e energy equation) adjoint advection terms were omitted in solvers/IncNavierStokesSolver/AdvectionTerms/AdjointAdvection.cpp, leading to wrong solutions of the adjoint (with scalar advection).

Proposed solution

The scalar adjoint advection terms (-U_j ds/dx_j, where s denotes the scalar) are added in solvers/IncNavierStokesSolver/AdvectionTerms/AdjointAdvection.cpp:181-191.

Implementation

The implementation is almost similar to solvers/IncNavierStokesSolver/AdvectionTerms/LinearisedAdvection.cpp except an if-statement is added in solvers/IncNavierStokesSolver/AdvectionTerms/AdjointAdvection.cpp:147 as a condition to compute the momentum (i < ndim) or scalar (i > ndim) adjoint advection.

Tests

The computation of adjoint and direct eigenvalues of a periodically heated thermal convection problem were added in solvers/IncNavierStokesSolver/Tests. The eigenvalues were validated with Hossain et. al. (2013) - Instabilities of natural convection in a periodically heated layer.

Specifically, 4 tests were added and they are:

1. ThermStab_dir.tst - Direct eigensolver (Modified Arnoldi)
2. ThermStab_dir_Ar.tst - Direct eigensolver (Arpack)
3. ThermStab_adj.tst - Adjoint eigensolver (Modified Arnoldi)
4. ThermStab_adj_Ar.tst - Adjoint eigensolver (Arpack)

These unit tests were tested on fenchurchstreet and my laptop which worked.

Suggested reviewers

Spencer Sherwin

Notes

Checklist

• Functions and classes, or changes to them, are documented.
• User guide/documentation is updated.
• Changelog is updated.
• Suitable tests added for new functionality.
• Contributed code is correctly formatted. (See the contributing guidelines).
• [ ] License added to any new files.
• No extraneous files have been added (e.g. compiler output or test data files).