Page 223 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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CONVECTION HEAT TRANSFER
This equation is simpler than that derived for a constant wall temperature and is limited
to the calculation of local non-dimensional wall temperatures (assuming T f is constant).
Therefore, the calculation of the Nusselt number on a wall subjected to a constant heat flux
is straightforward in any numerical method. However, in the Nusselt number calculation
for a surface subjected to a constant temperature, it is necessary to calculate the normal
temperature gradient. This calculation is simple if using a finite element discretization, in
which the normal gradient is equal to the boundary terms due to the discretization of the
second-order temperature terms, that is,
∂T ∂T ∂T ∂T
= n 1 + n 2 + n 3 (7.182)
∂n ∂x 1 ∂x 2 ∂x 3
where n 1 , n 2 and n 3 are the direction cosines of the surface normal. All the above discussed
quantities are local (on the surface nodes or elements). However, it is often necessary to
have an average Nusselt number for a heat transfer problem. The average Nusselt number
can be easily calculated by integrating the local Nusselt number over a length (in two
dimensions) or over a surface (in three dimensions). For example, in two dimensions,
nelem
1 1

Nu av = Nudl = Nu i dl i (7.183)
l l l
i=1
where l is the total length of the wall, i indicates a single incremental length of a one-
dimensional element on the wall on which the Nusselt number is calculated and n elem
indicates the total number of one-dimensional elements on the wall. If the length l in the
above equation is replaced by an area, then it can be directly applied to three-dimensional
problems. In order to use the above formula, the local Nusselt number over an incremental
length (dl i ) is assumed to be constant.

7.8.2 Drag calculation

The drag force is the resistance offered by a body that is equal to the force exerted by
the flow on the body at equilibrium conditions. The drag force arises from two different
sources. One is from the pressure p acting in the flow direction on the surface of the body
(form drag) and the second is due to the force caused by viscosity effects in the flow
direction. In general, the drag force is characterized by a drag coefficient, defined as
D
C d = 1 (7.184)
A f ρ a u 2
2 a
where D is the drag force, A f is the frontal area in the flow direction and the subscript a
indicates the free stream value. The drag force D contains the contributions from both the
influence of pressure and friction, that is,

D = D p + D f (7.185)
where D p is the pressure drag force and D f is the friction drag force in the flow direction.
The pressure drag, or form drag, is calculated from the nodal pressure values. For a two-
dimensional problem, the solid wall may be a curve or a line and the boundary elements on