![Sample our Engineering & Technology journals, sign in here to start your access, latest two full volumes FREE to you for 14 days]({"type":"image" , "src" : "/sda/5933/TOC-Subject-Sample-2021-Engineering-Tech.jpg"})
ABSTRACT
This work presents an alternative to the discretization of the Navier–Stokes equations using a finite volume method for hybrid unstructured grids with a staggered grid arrangement of variables. It has developed a numerical scheme, analogous to the element-based finite volume method, for the solution of 2-D incompressible fluid flow problems using several coupling strategies. All velocity components are stored at each face of the elements (pressure control volumes), following the usual procedure of staggering velocity and pressure. With this staggered arrangement, the balance of mass and momentum is satisfied, simultaneously, for the same set of variables, rendering numerical stability when compared to the nonstaggered arrangement.
Nomenclature
| A | = | neighboring coefficients |
| Ap | = | central coefficient |
| b | = | nodal source term integrated in the CV |
| B | = | generic source term |
| CV | = | control volume |
| dV | = | infinitesimal control volume |
| ds | = | cross-sectional area |
| E | = | time step factor multiplier |
| = | normal unit vector in the x direction | |
| I | = | identity matrix |
| = | normal unit vector in the y direction | |
| J | = | Jacobian |
| L | = | length of the square cavity |
| L2 | = | Euclidean norm |
| = | local mass flow | |
| = | mass inside the control volume at previous time level | |
| n | = | normal unity vector |
| nx | = | x-component of the normal unity vector |
| ny | = | y-component of the normal unity vector |
| Nb | = | number of nodal neighbors P and V of iv |
| Ne | = | number of elements of the mesh |
| Nv | = | number of vertices and faces of each element i |
| P | = | fluid pressure |
| P′ | = | pressure correction |
| Re | = | Reynolds number |
| s | = | mass flow sign |
| t | = | time |
| U | = | x-component of the velocity vector |
| = | lid-driven velocity (square cavity problem) | |
| V | = | y-component of the velocity vector |
| V | = | velocity vector |
| x, y | = | Cartesian coordinates |
| x | = | vector position |
| α, β | = | components of the metric tensor |
| Δ | = | distance between two points |
| ΔV | = | volume of the control volume |
| ε | = | maximum residue |
| η | = | local transverse coordinate at each face of a control volume CVV |
| λ | = | blending factor for the deferred correction interpolation scheme |
| μ | = | fluid viscosity |
| ξ | = | local parallel coordinate at each face of a control volume CVV |
| ρ | = | fluid density |
| Subscripts | = | |
| i | = | global index for the elements and nodal points for pressure |
| j | = | local index for each face of CVV |
| k | = | global index at each face of CVP and nodal points for velocity vector |
| ip | = | integration points |
| iv | = | global index of the vertices of the elements and control volumes |
| nb | = | nearest nodal points of the velocity vector |
| Superscripts | = | |
| o | = | value at the previous (old) time level |
| * | = | without the transient term |
| FO | = | first-order differencing scheme |
| P | = | pressure |
| SO | = | second-order differencing scheme |
| V | = | velocity |
Nomenclature
| A | = | neighboring coefficients |
| Ap | = | central coefficient |
| b | = | nodal source term integrated in the CV |
| B | = | generic source term |
| CV | = | control volume |
| dV | = | infinitesimal control volume |
| ds | = | cross-sectional area |
| E | = | time step factor multiplier |
| = | normal unit vector in the x direction | |
| I | = | identity matrix |
| = | normal unit vector in the y direction | |
| J | = | Jacobian |
| L | = | length of the square cavity |
| L2 | = | Euclidean norm |
| = | local mass flow | |
| = | mass inside the control volume at previous time level | |
| n | = | normal unity vector |
| nx | = | x-component of the normal unity vector |
| ny | = | y-component of the normal unity vector |
| Nb | = | number of nodal neighbors P and V of iv |
| Ne | = | number of elements of the mesh |
| Nv | = | number of vertices and faces of each element i |
| P | = | fluid pressure |
| P′ | = | pressure correction |
| Re | = | Reynolds number |
| s | = | mass flow sign |
| t | = | time |
| U | = | x-component of the velocity vector |
| = | lid-driven velocity (square cavity problem) | |
| V | = | y-component of the velocity vector |
| V | = | velocity vector |
| x, y | = | Cartesian coordinates |
| x | = | vector position |
| α, β | = | components of the metric tensor |
| Δ | = | distance between two points |
| ΔV | = | volume of the control volume |
| ε | = | maximum residue |
| η | = | local transverse coordinate at each face of a control volume CVV |
| λ | = | blending factor for the deferred correction interpolation scheme |
| μ | = | fluid viscosity |
| ξ | = | local parallel coordinate at each face of a control volume CVV |
| ρ | = | fluid density |
| Subscripts | = | |
| i | = | global index for the elements and nodal points for pressure |
| j | = | local index for each face of CVV |
| k | = | global index at each face of CVP and nodal points for velocity vector |
| ip | = | integration points |
| iv | = | global index of the vertices of the elements and control volumes |
| nb | = | nearest nodal points of the velocity vector |
| Superscripts | = | |
| o | = | value at the previous (old) time level |
| * | = | without the transient term |
| FO | = | first-order differencing scheme |
| P | = | pressure |
| SO | = | second-order differencing scheme |
| V | = | velocity |