FLOW AROUND A CYLINDER - istiarto

4.1 Governing equations
– 4.2 –
The flow model that is developed in this work is based on the approximate solution of the
time-averaged equations of motion and continuity for incompressible flows by using
finite-volume method. In the Cartesian coordinate system these equations read:
�u
�t
�v
�t
�w
�t
�u
�x
� �uu
�x
� �uv
�x
� �uw
�x
� �vu
�y
� �vv
�y
� �vw
�y
� �wu
�z
� �wv
�z
� �ww
�z
1 �p
� �
� �x
1 ��xx 1 ��yx 1 ��zx � � �
� �x � �y � �z � gx 1 �p 1 �� xy 1 ��yy 1 ��zy � � � � �
� �y � �x � �y � �z � gy 1 �p 1 �� xz 1 �� yz 1 ��zz � � � � �
� �z � �x � �y � �z � gz (4.1)
(4.2)
(4.3)
�v �w
� � � 0 (4.4)
�y �z
in which x, y, and z are Cartesian co-ordinates in the horizontal, transversal, and vertical,
respectively; u, v, and w are the corresponding (time-averaged) velocity components, p is
the (time-averaged) pressure, � is the mass density of water, g x , g y , g z are the x, y , z
components of the gravitational acceleration, and �ij‘s are the j direction components of
the shear stress acting on the surface normal to the i direction. These stresses are due to
the molecular viscosity and turbulent fluctuation. For flows having sufficiently high
Reynolds number, the viscous stresses are much smaller in comparison with those of the
turbulence and thus can be neglected. Using Boussinesq‘s eddy viscosity concept, these
stresses are proportional to the velocity gradients according to the following expressions
(see Rodi, 1984, p. 10):
� xx
� � �t 2 �u 2 � 3
�x k,
� yy
� � �t 2 �v 2
� 3 k,
�y
�zz � � �t2 �w 2
� 3 k,
�z
� xy
� � � yx
� xz
� � � zx
� yz
� � � zy
� � � ���v
�u��
t �� � ��,
���x
�y ��
� � � ���w
�u��
t �
���x
�z �� ,
� � � ���w
�v��
t �
��
��
�y �z��
��.
(4.5)
in which �t is the turbulent or eddy viscosity and k is the turbulent kinetic energy defined
as k � 1 2�u
�� u ���
v �� v ���
w ��w
���
where superscripts mean the fluctuating components.
Inserting the definitions in Eq. 4.5 into the momentum equations, Eqs. 4.1 to 4.3, one
obtains: