FLOW AROUND A CYLINDER - istiarto

– 5.41 –
Production of turbulent kinetic energy along cylinder boundaries
It has been observed that the computed turbulent kinetic-energy close to the cylinder is
below the measured values. This might be caused by a computational inaccuracy of its
production, due to an inherent difficulty related to the method of wall boundary
treatment. The computation of the production of turbulent kinetic energy along the
cylinder (wall) boundaries is done with a simplification, in which, among others, the
production due to the normal stress is neglected (see Eqs. 4.75a,b in Chapter 4). This
simplification is in conjunction with the wall function method applied to wall boundaries.
This method assumes that the velocity at nodes adjacent to the boundary is parallel to the
wall and is, between the wall and the adjacent node, distributed logarithmically. It has
been pointed out in the previous chapter that this is a rather rough approximation at the
cylinder boundaries. At such boundaries, the measured tangential and normal velocities,
notably in front of the cylinder, have the same order of magnitude. This is observed also
by the simulation (see Fig. 5.17a,e).
To render further insight into this problem, the computation of the production of turbulent
kinetic energy upstream and downstream of the cylinder is elaborated and presented in
Fig. 5.22. Shown in the figure are the computed tangential and normal velocity
components (Fig. 5.22a,b,c). Obviously, the normal velocities are not negligible.
Accordingly, neither the normal stresses are negligible compared to the tangential
components (Fig. 5.22d,e). The equation of the production of turbulent kinetic energy
written for wall boundaries is (see Eq. 4.75a in Chapter 4):
G � � nt
�
�Vt �n � �nn �Vn � �n
(4.75a)
where subscripts n and t are the normal and tangential directions, respectively. In the wall
function, the universal logarithmic velocity distribution is applied to the tangential
velocity. The tangential velocity gradient in Eq. 4.75a is obtained from this logarithmic
distribution. The production of turbulent kinetic energy due to the normal stress is simply
neglected. The production, G, is therefore computed by (see Eqs. 4.75b in Chapter 4):
G � � nt
�
�V t
�n
(4.75b)
which is plotted by solid lines in Fig. 5.22f,g. It was mentioned previously that this
production was not sufficient to generate the turbulent kinetic energy. The simulation
under-estimates the experimental values.
Now suppose that, in the absence of knowledge on the normal velocity distribution, we
assume a linear distribution of the normal velocities between the wall and the adjacent
node. We subsequently use this linear distribution to obtain the normal velocity gradient
and the normal stress in Eq. 4.75a. The production term thus obtained is shown by pointdashed
lines in Fig. 5.22f,g. Evidently the production increases notably in front of the
cylinder (Fig. 5.22f). There is also an increase, although not much, of the production of
turbulent kinetic energy downstream of the cylinder (Fig. 5.22g).