Turbulent Flow and Boundary Layer Theory: Selected Topics and Solved Problems

Turbulent flow will be the subject of research during this period and near future and hundreds of papers and articles are being published every year.

In this chapter, we emphasize on the physics side of turbulent flow with the engineering application, nature of turbulent, methods on analysis, diffusivity of turbulence, length scales in turbulent flows, energy and vorticity and their relation with length scale.

Diffusive and convective length scales and their relation with boundary thickness at the laminar and turbulent region and a comparison between them are presented in details.

The Navier – stokes equations describe the transport of momentum in a viscous fluid. For a laminar flow, these N.S.E. can be solved directly, often to a high degree of accuracy.

For turbulent flow, things are more complex. The equations describe the instantaneous velocity components u, v, and w at every point in the flow. However, the nature of turbulence is such that there are very strong variation in these quantities over small distance. The time over which fluctuations in velocity occurs are likewise very small.

In this chapter a brief explanation and derivation of the N.S.E. for turbulent flow which they called "Reynolds stress".

Prandtl mixing length theory also presented to solve the "Reynolds stress" related to a length scale and velocity gradient. In addition, the velocity profiles for turbulent flow described throughout an experimental variation of inner – outer and overlap layer laws.

Two major questions arise. First, how is the kinetic energy of turbulence maintained? Second, why are vorticity and vortex stretching so important to the study of turbulence? To help answer these questions. First, derive equations for the kinetic energy of the mean flow and that of turbulence.

In this chapter, the derivation of two – dim. K.E. was presented in full details and then mads assumption to find out the solution of the equation in simple way. The approximated velocity profile with strong pressure gradient and at the separation point are presented.

The study of hydraulic transients began with the investigation of the propagation of sound waves in air, the propagation in shallow water and flow of blood in arteries. However, none of these problems could solved rigorously until the development of elasticity and calculus and the solution of parties' differential equations.

In this chapter, a number of commonly used terms are defined, and a brief history of the development of the knowledge of hydraulic transients is presented. The basic water hammer equations for the change in pressure caused by an instantaneous change in flow velocity are then derived. A description of the propagation and reflection of waves produced by closing value at downstream and of a single pipeline is presented. This is followed by a discussion of the classification and causes of hydraulic transients.

Unsteady flow through closed conduits is described by the dynamic and continuity equations. The derivation of these equations is presented and methods available for their solution are discussed.

Methods for controlling transients flow using various devices available to reduce or to eliminate the undesirable transients. Boundary conditions for these devices will be developed, which are required for the analysis of a system by the method of characteristics.

Several exact solutions considered in literature notably the moving – boundary flows, stagnation flow, the rotating disk, convergent – wedge flow, and the flat plate with asymptotic suction – have hinted strongly at boundary – layer behavior. That is, at large Reynolds numbers the effect of viscosity becomes increasingly confined to narrow regions near solid walls. The computer models solutions also shewed this tendency at large Re to sweep the vorticity downstream and leave the flow far from the walls essentially irrational. Physically, this means that the rate of downstream convections is much larger than the rate of transverses viscous diffusion.

In this chapter, definitions related to the Boundary layer are described by a formula. A brief description and derivation in details of Boundary layer.

Theories, integral momentum equation of boundary layer implemented on flat – plate for laminar and turbulent flow, simulation solution for steady 2D flow, Blasius solution, Falkner – Skan wedge flow, and Thwaite's method.

The greatest number of practical applications the boundary layer is turbulent rather than laminar. Alternatively, more precisely, the bulk of the boundary layer is turbulent, with a thin laminar sublayer near the wall. Physically, turbulence acts to magnify enormously the local, instantaneous gradients of velocity, and thus is greatly augments the viscous stresses and acting in the fluid. The most important practical consequence of this is that the skin – friction for turbulent boundary layer are several orders of magnitude layer than the corresponding values for a laminar boundary layer at the same Reynolds number.

In this chapter, a brief description and derivation of turbulent boundary layer equation. The relation between stress (Reynolds and viscous stresses) and pressure gradients i.e. the relative magnitudes of the viscous and Reynolds stresses across a boundary layer. An estimation of laminar sub – layer due to neglecting the Reynolds stresses and the inner region of the turbulent layer by neglecting viscous term. The skin friction coefficient as a function of x and derived their relation with velocity profile along turbulent region.

Experimental evidence has shown that the transition process in a boundary layer starts locally at a number of points, which are more or less randomly distributed in both stream wise direction near the wall in region of finite stream wise extent at some distance from the leading edge. These transitions probably occur because of local instabilities of the mean basic flow to disturbances.

In this chapter, the transition from laminar to turbulent flow zone discussed physically and theoretically. The stability of flow and their relation with disturbance of flow presented in brief way.

The boundary layer velocity profile and all the related divination are derived by assuming that the momentum thickness will remain constant across the transition position.

Finally, the methods of boundary layer control to prevent separation in order to reduce and attain nigh lift described.