Page 59 - Computational Fluid Dynamics for Engineers

44 2. Conservation Equations

second, third and fourth terms denote the viscous forces per unit volume, and
they arise as a result of the different components of normal and shear stresses
shown in Fig. 2.1: the first subscript to the symbol a represents the direction of
the stress and the second the direction of the surface normal. By convention, an
outward normal stress acting on the fluid in the control volume is positive, and
the shear stresses are taken as positive on the faces furthest from the origin of
the coordinates. Thus a xy acts in the positive x direction on the visible (upper)
face perpendicular to the y axis; a corresponding shear stress acts in the negative
x direction on the invisible lower face perpendicular to the y axis.
Sometimes it is more convenient to write the viscous terms in the momentum
equations in tensor notation as 226
i * <- >

with z,j = 1,2,3 for three-dimensional flows: for example, i = 1, j = 1, 2,3 for
Eq. (2.2.2). For a constant density "Newtonian" viscous fluid, the normal viscous
stresses aij (i = j) and shear stresses aij (i ^ j) are obtained from the viscous
stress tensor given by
(dui duj\ . ^ .
=ft
^ {w j + di) (2 2 7)
--
According to Eq. (2.2.7), the normal viscous stress <J XX and the shear stresses
and in Eq. (2.2.2) are given by
a xy a xz
^ 9u (du dv\ f du dw\ oX
*xx = 2 / * ^ , ffastf = ^ ^ - + - j , ffx, = / ^ _ + _ j (2.2.8)
with similar expressions for the viscous stress tensor terms in Eqs. (2.2.3) and
(2.2.4).
In terms of Eq. (2.2.7), the Navier-Stokes equations can be simplified con-
siderably so that, for example, the x-momentum equation, Eq. (2.2.2) for a
Newtonian fluid becomes

Dt QOX
with similar expressions for the y- and ^-components obtained from Eqs. (2.2.3)
and (2.2.4). The resulting equations can be written in vector form as

2
^ = --Vp + vV V + f (2.2.10)
with V 2 denoting the Laplacian operator

d 2 d 2 d 2
„ 2
dx 2 dy 2 dz 2
The equation representing conservation of energy has a form similar to that
of the momentum equations (2.2.2) to (2.2.4). It is based on the first law of