### Introduction to the block-structured mesh

The block-structured mesh and related solution methods for transport equations on it are preferred by consideration of the fluid flow problems with high Reynolds and Grashof numbers and turbulence. The advection of heat and of species can be also easier computed on the structured mesh.

The structured mesh allows to achieve the required resolution in the boundary layers using much less mesh elements than with the unstructured mesh. The higher mesh resolution results in the application of the mesh cells with high aspect ratio.

The structured mesh is organized in numerical blocks. Each numerical block is a polygon with at least 4 corners which mark 4 block sides which are pairwise opposite to each other. The structured mesh inside of the block consists of two systems of the mesh lines. The number of the mesh elements at the opposite sides should be the same. Therefore the structured mesh in the block may be considered as a coordinates transformation, which maps the orthogonal equidistant mesh system into another not necessary equidistant and orthogonal mesh.

The transformed orthogonal mesh in the figure above defines a system of local coordinates associated with mesh lines. Each mesh line is considered as a local coordinate line. The discretization procedure is assembled in the generally curvilinear coordinate system associated with the given structured mesh.

The local coordinate system associated with the mesh in the figure is defined by the coordinate axes "xi" and "eta". Local coordinates are integers corresponding to the numbering of the mesh lines. The mesh elements are cells bounded by the adjacent mesh lines (or coordinate lines). The coordinate transformation maps the orthogonal mesh cell into the curvilinear cell as shown with red lines in the figure on the right side.

The control volumes (CV) used for discretization coincide with the mesh cells. Each control volume has 4 sides: West, East, North and South as shown in the figure for the selected mesh cell. Each of the CV sides separates the considered CV from neighbor. The 4 neighbor CV's are named too as CV-West, CV-East, CV-North and CV-South. The same "geographic" names will be used further also for 4 block sides.

All field values are computed and stored in the center of control volumes. The central node identified with letter "A" in the figure is one of such central nodes. Besides there are nodes located at the block boundaries and in the block corners. They are classified formally as central nodes of the corresponding 1D and 0D control volumes.

The nodes in 4 corners of the considered 2D block (letter B in the figure) are leaved always in their positions without respect to the block environment. Other rules are valid for nodes at the block boundary (positions C and D in the figure). The block boundary is indicated in the example in the figure with green line. The nodes of the block boundary remains at the middle position given by the block geometry everywhere at the block boundary without direct contact with the neighboring block. It may be the external boundary of the computational domain, symmetry axis or the interface with the external unstructured domain. Such unmoved nodes are located in the example figure at the south and north side of the block.

At the shared block boundary the nodal position is shifted towards position of the central node of the adjacent control volume in the neighbor block. The central node on the boundary of the neighbor block is shifted on his part towards the central node of the 2D control volume in the considered block. The shared block boundaries with shifted nodes are west and east block side in the example. The nodes displacement and the subsequent data exchange between the shifted nodes and nodes in their new locations assure the consistent treatment of the inner block boundary as if it would be one of the inner mesh lines.

The solution procedure on the structured mesh involves the SIP solver. Additionally the discretized enthalpy transport equation is solved using more general solvers applied also for the unstructured mesh. Exceptions are the discretized single species and chemical model (multiple species) transport equations which are solved only with one of solvers concepted for the unstructured mesh.

The iterative SIP solver is specialized for the quick initial residual reduction (or residual smoothing) for the matrix equation resulted from the discretization on the block-structured mesh. This advantage is especially useful for an iterative segregated solution of the nonlinear transport equations. The discretization on the block-structured mesh according to the Finite Volume method with non-staggered arrangement of the primary variables produces the diagonally occupied matrix of the linearized equations system for each transported value and for the pressure correction according to the SIMPLE method. The 5 diagonals occupied matrix arises from the discretization of the 2D transport problem.

The SIP solver solves the equations system block wise successive for all blocks. Within one inner solver iteration all blocks are processed. The inner solver iteration is executed for a block and then the modified nodal values in the last control volume at the inner block boundary are communicated to the neighbor block. After the inner iterations loop has finished, the residuals in all processed blocks should be reduced at least by the prescribed factor.

The neighboring blocks have the matched boundary with respect to the structured mesh from both sides from the considered shared block boundary. The boundary between the structured mesh domain and the unstructured mesh outside it is non-matched with respect to the structured mesh in the structured domain and the unstructured domain outside. The only exception is the case of the phase transition crystal-melt where structured and unstructured mesh lines should match exactly. All types of the boundaries are assured automatically by the automatic generator of the structured mesh and by the subsequent operations on it.