Notes on Computational Fluid Dynamics: General Principles

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4.2 Fixed value and ﬁxed gradient

In an equation for $\text{[math]}$ that is discretised to form a matrix equation $\text{[math]}$ , there are two terms that include properties interpolated to faces:

an advection term of the form $\text{[math]}$ which is discretised by Eq. (3.8 ) in extensive form (see Sec. 3.5 ) as $\text{[math]}$ ;
a Laplacian term of the form $\text{[math]}$ which is discretised by Eq. (3.2 ) in extensive form as $\text{[math]}$ .

The advection term requires the value $\text{[math]}$ and Laplacian term requires the surface normal gradient $\text{[math]}$ at faces. When a face is part of a boundary patch, any gradient $\text{[math]}$ and/or value $\text{[math]}$ must be speciﬁed through boundary conditions.

The ﬁxed value, or Dirichlet condition,¹ is the ﬁrst type of boundary condition, where the boundary value $\text{[math]}$ is speciﬁed. For example, at an inlet patch, we might specify a temperature $\text{[math]}$ 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 $\text{[math]}$ is speciﬁed (where $\text{[math]}$ ). In many cases, the applied normal gradient is zero, which is a common condition applied to many ﬁelds, including $\text{[math]}$ , at an outlet patch, i.e. where the ﬂuid ﬂows out of the domain.

$PICT\relax \special {t4ht=$

When an advection term is discretised, a ﬁxed value condition is applied by substituting the face value $\text{[math]}$ with the patch value $\text{[math]}$ , i.e. setting $\text{[math]}$ . When a ﬁxed gradient $\text{[math]}$ is speciﬁed, the face value is expressed as follows, where $\text{[math]}$ is the value in the cell adjacent to each face:

r f = P + --n--b: C \relax \special {t4ht=

(4.2)

When a Laplacian term is discretised, a ﬁxed gradient boundary condition is applied by substituting the face normal gradient $\text{[math]}$ with the patch face normal gradient $\text{[math]}$ , i.e. setting $\text{[math]}$ . When a ﬁxed value $\text{[math]}$ is speciﬁed, the face normal gradient is expressed by

rn f = C ( b P): \relax \special {t4ht=

(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