viscous incompressible flow

viscous incompressible flow
Being highly non-linear due to the convective acceleration terms, the Navier-Stokes equations are difficult to handle in a physical situation. Moreover, there are no general analytical schemes for solving the nonlinear partial differential equations. However, there are few applications where the convective acceleration vanishes due to the nature of the geometry of the flow system. So, the exact solutions are often possible. Since, the Navier-Stokes equations are applicable to laminar and turbulent flow s, the complication again arise due to fluctuations in velocity components for turbulent flow. So, these exact solutions are referred to laminar flows for which the velocity is independent of time (steady flow) or dependent on time (unsteady flow) in a well-defined manner. The solutions to these categories of the flow field can be applied to the internal and external flows . The flows that are bounded by walls are called as internal flows while the external flows are unconfined and free to expand. The classical example of internal flow is the pipe/duct flow while the flow over a flat plate is considered as external flow. Few classical cases of flow fields will be discussed in this module pertaining to internal and external flows. 
Laminar and Turbulent Flows 
The fluid flow in a duct may have three characteristics denoted as laminar, turbulent and transitional. The curves shown in Fig. 5.1.1, represents the x- component of the velocity as a function of time at a point ‘A' in the flow. For laminar flow, there is one component of velocity and random component of velocity normal to the axis becomes predominant for turbulent flows i.e. . When the flow is laminar, there are occasional disturbances that damps out quickly. The flow Reynolds number plays a vital role in deciding this characteristic. Initially, the flow may start with laminar at moderate Reynolds number. With subsequent increase in Reynolds number, the orderly flow pattern is lost and fluctuations become more predominant. When the Reynolds number crosses some limiting value, the flow is characterized as turbulent. The changeover phase is called as transition to turbulence . Further, if the Reynolds number is decreased from turbulent region, then flow may come back to the laminar state. This phenomenon is known as relaminarization 
Fig. 5.1.1: Time dependent fluid velocity at a point.
The primary parameter affecting the transition is the Reynolds number defined as, where, U is the average stream velocity and L is the characteristics length/width. The flow regimes may be characterized for the following approximate ranges; 
Fully Developed Flow 
The fully developed steady flow in a pipe may be driven by gravity and /or pressure forces. If the pipe is held horizontal, gravity has no effect except for variation in hydrostatic pressure. The pressure difference between the two sections of the pipe, essentially drives the flow while the viscous effects provides the restraining force that exactly balances the pressure forces. This leads to the fluid moving with constant velocity (no acceleration) through the pipe. If the viscous forces are absent, then pressure will remain constant throughout except for hydrostatic variation. 
In an internal flow through a long duct is shown in Fig. 5.1.2. There is an entrance region where the inviscid upstream flow converges and enters the tube. The viscous boundary layer grows downstream, retards the axial flow at the wall and accelerates the core flow in the center by maintaining the same flow rate. 
Fig. 5.1.2: Velocity profile and pressure changes in a duct flow.
At a finite distance from entrance, the boundary layers form top and bottom wall merge as shown in Fig. 5.1.2 and the inviscid core disappears, thereby making the flow entirely viscous. The axial velocity adjusts slightly till the entrance length is reached and the velocity profile no longer changes in x and only. At this stage, the flow is said to be fully-developed for which the velocity profile and wall shear remains constant. Irrespective of laminar or turbulent flow, the pressure drops linearly with x. The typical velocity and temperature profile for laminar fully developed flow in a pipe is shown in Fig. 5.1.2. The most accepted correlations for entrance length in a flow through pipe of diameter (d) , are given below; 

Laminar and Turbulent Shear 
In the absence of thermal interaction, one needs to solve continuity and momentum equation to obtain pressure and velocity fields. If the density and viscosity of the fluids is assumed to be constant, then the equations take the following form; 

This equation is satisfied for laminar as well as turbulent flows and needs to be solved subjected to no-slip condition at the wall with known inlet/exit conditions. In the case of laminar flows, there are no random fluctuations and the shear stress terms are associated with the velocity gradients terms such as, in x- direction. For turbulent flows, velocity and pressure varies rapidly randomly as a function of space and time as shown in Fig. 5.1.3. 
Fig. 5.1.3: Mean and fluctuating turbulent velocity and pressure.
One way to approach such problems is to define the mean/time averaged turbulent variables. The time mean of a turbulent velocity is defined by, 

where, T is the averaging period taken as sufficiently longer than the period of fluctuations. If the fluctuation is taken as the deviation from its average value, then it leads to zero mean value. However, the mean squared of fluctuation is not zero and thus is the measure of turbulent intensity 
In order to calculate the shear stresses in turbulent flow, it is necessary to know the fluctuating components of velocity. So, the Reynolds time-averaging concept is introduced where the velocity components and pressure are split into mean and fluctuating components; 
Substitute Eq. (5.1.6) in continuity equation (Eq. 5.1.3) and take time mean of each equation; 
Let us consider the first term of Eq. (4.1.7), 
Considering the similar analogy for other terms, Eq. (5.1.7) is written as, 
This equation is very much similar with the continuity equation for laminar flow except the fact that the velocity components are replaced with the mean values of velocity components of turbulent flow. The momentum equation in x- direction takes the following form; 
The terms in RHS of Eq. (5.1.3) have same dimensions as that of stress and called as turbulent stresses . For viscous flow in ducts and boundary layer flows, it has been observed that the stress terms associated with the ydirection (i.e. normal to the wall) is more dominant. So, necessary approximation can be made by neglecting other components of turbulent stresses and simplified expression may be obtained for Eq. (5.1.10). 
It may be noted that are zero for laminar flows while the stress terms is positive for turbulent stresses. Hence the shear stresses in turbulent flow are always higher than laminar flow. The terms in the form of are also called as Reynolds stresses .
Turbulent velocity profile 
A typical comparison of laminar and turbulent velocity profiles for wall turbulent flows, are shown in Fig. 5.1.4(a-b). The nature of the profile is parabolic in the case of laminar flow and the same trend is seen in the case of turbulent flow at the wall. The typical measurements across a turbulent flow near the wall have three distinct zones as shown in Fig. 5.1.4(c). The outer layer is of two or three order magnitudes greater than the wall layer and vice versa. Hence, the different sub-layers of Eq. (5.1.11) may be defined as follows; 
•  Wall layer (laminar shear dominates) 
•  Outer layer (turbulent shear dominates) 
•  Overlap layer (both types of shear are important) 
Fig 5.1.4: Velocity and shear layer distribution: (a) velocity profile in laminar flow; (b) velocity profile in turbulent flow; (c) shear layer in a turbulent flow. 
In a typical turbulent flow, let the wall shear stress, thickness of outer layer and velocity at the edge of the outer layer be , respectively. Then the velocity profiles (u) for different zones may be obtained from the empirical relations using dimensional analysis. 
Wall layer : In this region, it is approximated that u is independent of shear layer thickness so that the following empirical relation holds good. 
Eq. (5.1.12) is known as the law of wall and the quantity is called as friction velocity. It should not be confused with flow velocity. 
Outer layer : The velocity profile in the outer layer is approximated as the deviation from the free stream velocity and represented by an equation called as velocity-defect law 
Overlap layer : Most of the experimental data show the very good validation of wall law and velocity defect law in the respective regions. An intermediate layer may be obtained when the velocity profiles described by Eqs. (5.1.12 & 5.1.13) overlap smoothly. It is shown that empirically that the overlap layer varies logarithmically with y (Eq. (5.1.14). This particular layer is known as overlap layer