## 4.2Fixed value and ﬁxed gradient

In an equation for that is discretised to form a matrix equation , there are two terms that include properties interpolated to faces:

• an advection term of the form which is discretised by Eq. (3.8 ) in extensive form (see Sec. 3.5 ) as ;
• a Laplacian term of the form which is discretised by Eq. (3.2 ) in extensive form as .

The advection term requires the value and Laplacian term requires the surface normal gradient at faces. When a face is part of a boundary patch, any gradient and/or value must be speciﬁed through boundary conditions.

The ﬁxed value, or Dirichlet condition,1 is the ﬁrst type of boundary condition, where the boundary value is speciﬁed. For example, at an inlet patch, we might specify a temperature K of the ﬂuid ﬂowing into the domain.

The ﬁxed gradient, or Neumann condition,2 is the second type, in which the gradient normal to the boundary is speciﬁed (where ). In many cases, the applied normal gradient is zero, which is a common condition applied to many ﬁelds, including , at an outlet patch, i.e. where the ﬂuid ﬂows out of the domain. When an advection term is discretised, a ﬁxed value condition is applied by substituting the face value with the patch value , i.e. setting . When a ﬁxed gradient is speciﬁed, the face value is expressed as follows, where is the value in the cell adjacent to each face: (4.2)
When a Laplacian term is discretised, a ﬁxed gradient boundary condition is applied by substituting the face normal gradient with the patch face normal gradient , i.e. setting . When a ﬁxed value is speciﬁed, the face normal gradient is expressed by (4.3)

1Peter Dirichlet, Über einen neuen Ausdruck zur Bestimmung der Dictigkeit einerunendlich dünnen Kugelschale, wenn der Werth des Potentials derselben in jedem Punkte ihrer Oberﬂäche gegeben ist, 1850.
2after Carl Neumann 1832-1925.

Notes on CFD: General Principles - 4.2 Fixed value and ﬁxed gradient 