Fluid Dynamics and the Navier-Stokes Equations

The subject of fluid dynamics deals with the motion of liquids and gases, which when studied macroscopically, appear to be continuous in structure. All the variables are considered to be continuous functions of the spatial coordinates and time. The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the flow of fluids. For example: they model weather or the movement of air in the atmosphere, ocean currents, water flow in a pipe, as well as many other fluid flow phenomena. The Navier-Stokes equations for irrotational flow (when

x u = 0) are:

(3)

where u = velocity vector field,

= thermodynamic internal energy, p = pressure, T = temperature,

= density,

= viscosity, K_H = heat conduction coefficient, and F = external force per unit mass.

The Navier-Stokes equations are time-dependent and consist of a continuity equation for conservation of mass, three conservation of momentum equations and a conservation of energy equation. There are four independent variables in the equation - the x, y, and z spatial coordinates, and the time t; six dependent variables - the pressure p, density

, temperature T, and three components of the velocity vector u. Together with the equation of state such as the ideal gas law - p V = n R T, the six equations are just enough to determine the six dependent variables. In general, all of the dependent variables are functions of all four independent variables. Usually, the Navier-Stokes equations are too complicated to be solved in a closed form. However, in some special cases the equations can be simplified and may admit analytical solutions.

Incompressible fluid - In fluid dynamics, an incompressible fluid is a fluid whose density is constant. It is the same throughout space and it does not change through time. According to the continuity equation, it also implies u = 0. It is an idealization used to simplify analysis. In reality, all fluids are compressible to some extent.
Inviscid or Stokes flow - Viscous problems are those in which fluid friction have significant effects on the solution. Problems for which friction can safely be neglected are called inviscid. The Reynolds number (R=(u_sL)/, where u_s is the mean fluid velocity, and L is the characteristic length, e.g., the cross-section of the pipe) can be used to evaluate whether viscous or inviscid equations are appropriate to the problem. High Reynolds numbers indicate that the inertial forces are more significant than the viscous forces. However, even in high Reynolds number regimes certain problems require that viscosity be included. In particular, problems calculating net forces on bodies (such as the wings on aircraft) should use viscous equations. Stokes flow occurs at very low Reynold's numbers, such that inertial forces can be neglected compared to viscous forces.
Steady flow - Another simplification of the equations is to set all changes of fluid properties with time to zero. This is called steady flow, and is applicable to a large class of problems, such as lift and drag on a wing or flow through a pipe.
Boussinesq approximation - In fluid dynamics, the Boussinesq approximation is used in the field of buoyancy-driven flow. It states that density differences are sufficiently small to be neglected, except where they appear in terms multiplied by g, the acceleration due to gravity. The essence of the Boussinesq approximation is that the difference in inertia is negligible but gravity is sufficiently strong to make the specific weight appreciably different between the two fluids. Boussinesq flows are common in nature (such as atmospheric fronts, oceanic circulation, downhill winds), industry (dense gas dispersion, fume cupboard ventilation), and the built environment (natural ventilation, central heating). The approximation is extremely accurate for many such flows, and makes the mathematics and physics simpler.

Laminar vs turbulent flow - Turbulence is flow dominated by recirculation, eddies, and apparent randomness (see Figure 01). Flow in which turbulence is not exhibited is called laminar (see Figure 02). It is believed that turbulent flows obey the Navier-Stokes equations. However, the flow is so complex that it is not possible to solve turbulent problems from first principles with the computational tools available today or likely to be available in the near future. Turbulence is

		instead modeled using one of a number of turbulence models and coupled with a flow solver that assumes laminar flow outside a turbulent region. Turbulence usually occurs below a Reynold's numbers of 3000. It causes increased energy loss (as heat), more drag (on the moving body), and generates sound wave (noise).
Figure 01 Turbulent Flow [view large image]	Figure 02 Laminar Flow [view large image]	Modern vehicle and aircraft designs always try to minimize the turbulence by adopting a smooth surface and streamlined contour.

The steady flow in a pipe is a very simple solvable case. Since the left-hand side of Eq.(2) vanishes for steady flow, and there is no external force, it can be written in cylinderical coordinates as:

0 = - dp / dz +

(1/r) d (r d v / dr) / dr ---------- (4)

where v denotes the velocity field in the z direction as shown in Figure 02. For a constant pressure drop (per unit length)
d p / dz is a negative constant, the solution becomes:

v (r) = v_m ( 1 - r² / R² ) ---------- (5)

where v_m = (R² / 4

) (- dp / dz) ---------- (6)
is the velocity at the centre, and R is the radius of the pipe. The boundary conditions are v = v_m at r = 0 and v = 0 at r = R.

Eq.(5) shows that the constant velocity contours are arranged in layers of concentric cylinders with the maximum velocity v_m at the center. There is no mixing of layers. This is one way to define laminar flow.
The definition of v_m in Eq.(6) shows that the flow is driven by the pressure gradient ( - dp / dz). Viscosity is one of the factors in determining the velocity of the flow, which becomes slower for more viscous fluid. The radius of the pipe R also has an influence on the flow, which becomes slower in smaller pipe. The flow comes to a halt with v_m = 0 at the limit of either ( - dp / dz) = 0 or ~ infinity.
The flow comes to a stop at the wall. This is the consequence of an assumption that the frictional force between the fluid and the wall is infinite. The effect is similar to the thin film of liquid sticking to a solid surface. If this is not the case, then v_m would take the more general form:

v_m = (R² / 4 ) (- dp / dz) + v₀ ---------- (7)

where v₀ is the velocity of the flow at the wall. The solution for the velocity field in Eq.(5) becomes:

v (r) = v_m ( 1 - r² / R² ) + (r / R)² v₀---------- (8)

In case when the pressure gradient is absent, i.e., (- dp /dz) = 0 in Eqs.(4) and (7). The above solution is reduced to v(r) = v_m = v₀, that is, the flow has an uniform velocity similar to the case shown in Figure 03 for the uniform flow of inviscid fluid. But in this example, the viscosity term disappears in Eqs.(4) and (7) because (- dp /dz) = 0; and thus has no influence on the flow. It seems that the flow can last forever without a driving force. In reality, it would finally come to a halt by the dissipative effects and the discharge at the end of the pipe (in this example an infinitely long pipe is assumed).

Figure 03 Uniform Flow [view large image]

	Instead of maintaining the pressure gradient by a pump, the flow can be driven by gravity if it is tilted by an angle A with respect to the horizon (Figure 04). Then (- dp / dz) = g sin(A) ---------- (9) where is the density of the fluid, g = 980 cm/sec² is the acceleration of gravity, and
Figure 04 Flow by Gravity Feed [view large image]	sin(A) is related to the slant of the pipe. Such method is a convenient way to transport liquid, but it does not work for gas since its density is about 1000 times lower.

Fluid Dynamics and the Navier-Stokes Equations

Figure 01 Turbulent Flow [view large image]

Figure 02 Laminar Flow [view large image]

Figure 03 Uniform Flow [view large image]

Figure 04 Flow by Gravity Feed [view large image]