External flow over immersed bodies If a body is immersed in a flow, we call it an external flow. Some important external flows include airplanes, motor vehicles, and flow around buildings. Typical quantities of interest are lift and drag acting on these objects. Often flow modeling is used to determine the flow fields in a wind tunnel or water tank. External flows involving air are typically termed aerodynamics.

Developing a good understanding of external flow is important in the design of many engineering systems such as aircraft, automobiles, buildings, ships, submarines, and all kinds of turbines. Late-model cars, for example, have been designed with particular emphasis on aerodynamics. This has resulted in significant reductions in fuel consumption and noise, and considerable improvement in handling. Applications

The velocity of the fluid approaching a body is called the free- stream velocity and is denoted by V. It is also denoted by u ∞ or U ∞ when the flow is aligned with the x-axis. Types of External Flows: Two-Dimensional: infinitely long and of constant cross-sectional size and shape. Axisymmetric: formed by rotating their cross- sectional shape about the axis of symmetry. Three-Dimensional: may or may not possess a line of symmetry.

Types of External Flows: Flow over bodies can also be classified as incompressible flows (e.g., flows over automobiles, submarines, and buildings) and compressible flows (e.g., flows over high-speed aircraft, rockets, and missiles). Compressibility effects are negligible at low velocities (flows with Ma < 0.3), and such flows can be treated as incompressible. Bodies subjected to fluid flow are classified as being streamlined or blunt, depending on their overall shape. A body is said to be streamlined if a conscious effort is made to align its shape with the anticipated streamlines in the flow. Streamlined bodies such as race cars and airplanes appear to be contoured and sleek. Otherwise, a body (such as a building) tends to block the flow and is said to be bluff or blunt. Usually it is much easier to force a streamlined body through a fluid, and thus streamlining has been of great importance in the design of vehicles and airplanes Types of External Flows: The shape of a body affects the flow characteristics.

A body interacts with the surrounding fluid through pressure and shear stresses. Lift and Drag Concepts

Of course, to carry out the integrations and determine the lift and drag, we must know the body shape(i.e., theta as a function of location along the body) and the distribution of and p along the surface.

Lift coefficients and drag coefficients are dimensionless forms of lift and drag.

SKIN FRICTION AND PRESSURE DRAG The part of drag that is due directly to wall shear stress tw is called the skin friction drag (or just friction drag FD, friction) since it is caused by frictional effects, and the part that is due directly to pressure P is called the pressure drag (also called the form drag because of its strong dependence on the form or shape of the body). The friction and pressure drag coefficients are defined as

Reducing Drag by Streamlining

Streamlining should be considered only for blunt bodies that are subjected to high-velocity fluid flow (and thus high Reynolds numbers) for which flow separation is a real possibility. Streamlining has the added benefit of reducing vibration and noise.

Flow Separation At sufficiently high velocities, the fluid stream detaches itself from the surface of the body. This is called flow separation (Fig. 11–13). Flow can separate from a surface even if it is fully submerged in a liquid or immersed in a gas (Fig. 11–14).

When a fluid separates from a body, it forms a separated region between the body and the fluid stream. This low-pressure region behind the body where recirculating and backflows occur is called the separated region. The larger the separated region, the larger the pressure drag. The effects of flow separation are felt far downstream in the form of reduced velocity (relative to the upstream velocity). The region of flow trailing the body where the effects of the body on velocity are felt is called the wake

The occurrence of separation is not limited to blunt bodies. Complete separation over the entire back surface may also occur on a streamlined body such as an airplane wing at a sufficiently large angle of attack (larger than about 15° for most airfoils), which is the angle the incoming fluid stream makes with the chord (the line that connects the nose and the end) of the wing. Flow separation on the top surface of a wing reduces lift drastically and may cause the airplane to stall. Stalling has been blamed for many airplane accidents and loss of efficiencies in turbomachinery

ice formation on airplane wings An important consequence of flow separation is vortex shedding

DRAG COEFFICIENTS OF COMMON GEOMETRIES

which is known as Stokes law,

External Flows: Flow Past slender Objects Flat Plate Flow: Low Reynolds Number: Re = 0.1 Medium Reynolds Number: Re = 10 Large Reynolds Number: Re = 10 5 Large Boundary Layer Thin Boundary Layer

Boundary layer thickness is so small that pressure field is solved considering the whole flow to be inviscid and then this pressure distribution is considered as driving force of the flow in the boundary layer.

External Flows: Flow Past bluff Objects Symmetric Separation Wake

Solution

Displacement thickness The boundary layer displacement thickness is defined in terms of volumetric flowrate.

The boundary layer momentum thickness is defined in terms of momentum flux. Momentum thickness

The Navier–Stokes equations can be simplified for boundary layer flow analysis. Continuity eqn X momentum eqn Blasius solution (Laminar Boundary Layer) Boundary conditions Navier Stoke’s eqn

Similarity variable converts it to Blasius solution 3 rd order ordinary differential equation

Von Karman’s Momentum-Integral Boundary Layer Equation for a Flat Plate

Drag in terms of momentum thickness Approximate velocity profiles are used in the momentum integral equation.

Assumed Laminar boundary layer velocity distribution Find the following quantities

Turbulent boundary layer on a flat plate Transition takes place in the range

The shear force for turbulent boundary layer flow is considerably greater than it is for laminar boundary layer flow in a flat plate because of random mixing of finite sized fluid particles. There are no exact solutions available for turbulent boundary layer flows as there is no precise expression for the shear stress in turbulent flow

Turbulent Laminar

Effects of Pressure Gradient The free-stream velocity on a curved surface is not constant. If there were no viscosity, there would be no pressure or friction drag on a cylinder.

Viscous effects within the boundary layer cause boundary layer separation.