CE6303 MF Notes, MECHANICS OF FLUIDS Lecture Handwritten Notes – CIVIL 3rd SEM Anna University

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 flows, 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 ˆ

component of velocity V u i and random component of velocity normal to the axis ˆ ˆ ˆ

becomes predominant for turbulent flows i.e. V u i v j w k . 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.