Fundamental equations
2.2 THE FUNDAMENTAL EQUATIONS .1 Introduction
2.2.3 The momentum equation The Navier-Stokes equation
Newton's law of motion is used as a basis for developing the momentum equation for a control volume:
X ^ = —[M XV\ = -:^(\^ypVdVolume) + l^^pVVdArea (2.6) where CV and CS refer to the control volume and control surface, respectively. Basically, it
states that the change of momentum equals the sum of all forces applied to the control volume.
The forces acting on the control volume are (1) the surface forces (i.e. pressure and shear forces) acting on the control surface and (2) the volume force (i.e. gravity) applied at the centre of mass of the control volume. For an infinitesimal small volume the momentum equation applied to the /-component of the vector equation is:
where D/Dt is the absolute differential (or absolute derivative, see Appendix A1.3), Vj is the velocity component in the /-direction, F^^i is the resultant of the volume forces (per unit vol- ume) and (Tij is the stress tensor (see notes below). The subscripts / andy refer to the Cartesian co-ordinate components (e.g. x, y).
If the volume forces F^oi derives from a potential U, they can be rewritten as: F^QX = - g r a d U (e.g. gravity force F^^i = —grad (gz)). Further, for a Newtonian fluid the stress forces are (1) the pressure forces and (2) the resultant of the viscous forces on the control volume. Hence for a Newtonian fluid and for volume force deriving from a potential, the momentum equation becomes:
of p X v]
Dt = P^voi ~ g^ad P + F , (2.8)
where P is the pressure and F^^^c is the resultant of the viscous forces (per unit volume) on the control volume.
In Cartesian co-ordinates (x, y, z):
J ' ^\
dt
dt
dt
- J J
= P^voL -
+ 1^;
+ 1^;
dx PF. vol, V J
= PKOL -
dp_
dx dP dy dP dz
+ F
^ VISC^
+ F
+ F
^ VISC^
(2.9a)
(2.9b)
(2.9c) where the subscript^* refers to the Cartesian co-ordinate components (i.e.y = x, y, z). In equation (2.9), the term on the left side is the sum of the momentum accumulation d(pV)/dt plus the momen- tum flux Vd(pV)/dx. The left term is the sum of the forces acting on the control volume: body force (or volume force) acting on the mass as a whole and surface forces acting at the control surface.
For an incompressible flow (i.e. p = constant), for a Newtonian fluid and assuming that the viscosity is constant over the control volume, the motion equation becomes:
3/ J ^ J
dK.
dt r ' dx
^
\
dt J ^^7 J
17 3 ^ ^ F
J7 ^ ^ ^ C
(2.10a)
(2.10b)
(2.10c) where p, the fluid density, is assumed constant in time and space. Equation (2.10) is often called the Navier-Stokes equation.
Considering a two-dimensional flow in the (x, y) plane and for gravity forces, the Navier- Stokes equation becomes:
^ ' 3 ; ' dx ' dy -ps dx X (2.11a)
2.2 The fundamental equations 13
dK,
dK,
dK,nir*"'!^*'''!;
dy dy (2.11b)where z is aligned along the vertical direction and positive upward. Note that the x- andy-directions are perpendicular to each other and they are independent of (and not necessarily orthogonal to) the vertical direction.
Notes
1. For gravity force the volume force potential t/is:
where g is the gravity acceleration vector and x = (x, y, z),z being the vertical direction positive upward. It yields that the gravity force vector equals:
7^ vol = -grad(gz)
2. A Newtonian fluid is characterized by a linear relation between the magnitude of shear stress r and the rate of deformation dVldy (equation (1.1)), and the stress tensor is:
atj = -P8ij + Tij
where P is the static pressure, r^ is the shear stress component of the /-momentum transported in the 7-direction, dy is the identity matrix element: 8^ = 1 and 8^ = 0 (for / different fromy).
1
^y = il
y^ dxj dx, I J 3. The vector of the viscous forces is:and e = d i v F = y ^ ^ , dx.
I S T , , -* vise, ^ ^ ^ ' I Z ^ -\
For an incompressible flow the continuity equation gives: e = div F = 0. And the viscous force per unit volume becomes:
^visc, 2^f^ d'v^
J 3*y 3^7
where fi is the dynamic viscosity of the fluid. Substituting this into equation (2.11), yields:
dt "^ dx ' ' dy' - - ^ ^ — - — " -dx dx 7 ^^7 3^7 (2.12a) dV^ dV^ dV^^
—^ + v —- + v —-
dt ^ dx ' dy
dz dP d'K,
dy dy , 3x, dx (2.12b)
7 " 7
Equation (2.12) is the ongzTza/Navier-Stokes equation.
4. Equation (2.12) was first derived by Navier in 1822 and Poisson in 1829 by an entirely different method. Equations (2.10)-(2.12) were derived later in a manner similar as above by Barre de Saint-Venant in 1843 and Stokes in 1845.
5. Louis Navier (1785-1835) was a French engineer who not only primarily designed bridge but also extended Euler's equations of motion. Simeon Denis Poisson (1781-1840) was a French mathematician and scientist. He developed the theory of elasticity, a theory of electricity and a theory of magnetism. Adhemar Jean Claude Barre de Saint-Venant (1797-1886), French engin- eer, developed the equations of motion of a fluid particle in terms of the shear and normal forces exerted on it. George Gabriel Stokes (1819-1903), British mathematician and physicist, is known for his research in hydrodynamics and a study of elasticity.
Application
Considering an open channel flow in a rectangular channel (Fig. 2.2), we assume a one-dimensional flow, with uniform velocity distribution, a constant channel slope 6 and a constant channel width B.
The Navier-Stokes equation in the ^-direction is:
dv . dv + V dt ds
-Pg ds ds "^•
where Fis the velocity along a streamline. Integrating the Navier-Stokes equation over the control vol- ume (Fig. 2.2), the forces acting on the control volume shown in Fig. 2.2 in the ^--direction are:
J
' dz —pg— = +pgA As sin 6 Volume force (i.e. weight)CV He
*dy
I - — = -pgd MB cos 6 Pressure force (assuming hydrostatic pressure distribution) dP
•'CV dy
f ^vis ~ "''"o^w ^^ Friction force (i.e. boundary shear) Jcv
where A is the cross-sectional area (i.e. A = Bd for a rectangular channel), d is the flow depth, A^ is the length of the control volume, TQ is the average bottom shear stress and P^ is the wetted perimeter.
Channel cross-sections
W
v»
D Fig. 2.2 Control volume for an openchannel flow.