Flow Over Immersed Bodies_A

Fluid Flow

It is characterized by a freely growing boundary layer surrounded by an outer flow region that involves small velocity and temperature gradients.

There are 2 types of fluid flow:

1. External Flow   ↠   Fluid flow when the object is completely surrounded by the fluid.
2. Internal Flow   ↠   Flow which are bounded by walls or fluid interfaces.

Consider 2 other flows:

• Boundary Layer Flow   ↠   High Reynold's number flow around streamlines bodies without flow separation.
• Bluff Body Flow   ↠   Flow around bluff body with flow separation.

Testing  ↠  Wind tunnel testing of model airplanes, building.

Three general body characteristics:

1. 2D Bodies   ↠   Infinitely long and of constant cross-sectional size and shape.
2. Axisymmetric Bodies   ↠   Formed by rotating their cross-sectional shape about the axis of symmetry.
3. 3D Bodies   ↠   May (or may not) possess line (or plane) of symmetry.

Body shape may depend on whether the body is streamlined or blunt. It is easy to force a streamlined body through a fluid than blunt body at same velocity.

1. Streamlined Bodies   ↠   Airfoils, racing cars have little effect on the surrounding fluid.
2. Blunt Bodies   ↠   Parachutes, buildings have effect on fluid.

Lift and Drag

Lift is defined as, "Resultant shear and pressure forces acting in the direction of the upstream velocity". Drag is defined as, "Resultant shear and drag forces acting normal to the upstream velocity".

• Froud Number   ↠   use for open channel flow in place of Reynold's number. But for modelling for any body, use Reynold's number.
• The characteristics of the flow depend very strongly on size, orientation, speed and fluid properties for a given shaped object.
• Reynold's Number  =  function (External flow, lift, drag).
• If Re > 100   ↠   flow is dominated by Inertial effects.
• If Re < 1    ↠    flow is dominated by Viscous effects.

Ques: Why flow separated?

Ans: With the increase in Reynold's number, fluid inertia becomes more important such that it cannot follow the curved path around to the rear of the body.

• At high Reynold's number, if thickness of boundary layer << diameter   ↠   Result in irregular and unsteady wake region.

Prandtl Solution to NSE

Hydrodynamics evolved from Euler's Equation of motion for an inviscid fluid. But in NSE there is not drag related terms, so engineers developed their own empirical art of hydraulics.

• Mathematical difficulties in solving these equations.

Prandtl suggest that Navier-Stokes equations can be solved by using:

1. Boundary Layer
2. Above the boundary layer (covering the rest of flow)

NSE Solving Concepts

Only in the boundary layer (close to solid boundary), viscosity effects are important. For E.g.: The flow past an object can be treated as a combination of viscous flow (in boundary layer) and inviscid flow elsewhere.

• Inside Boundary layer   ↠  Friction is significant and across the width velocity increases from zero to max. which Inviscid Flow Theory predicts.
• Outside Boundary layer   ↠   Velocity gradient perpendicular to the flow are relatively small and fluid acts as inviscid, viscosity is not zero.   

Inviscid or Potential Flow Theory

The theory suggests that, "Regions outside the boundary layer, effects of viscosity is negligible and flow may be treated as inviscid".

• Boundary layer is Laminar from the leading edge for a short period.
• Transition occurs over a region of plate.
• Transition region extends downstream to the location where the boundary layer flow becomes completely turbulent.
• Turbulent layer growth is more rapid that that of laminar layer.
• Boundary Layer Tripping   ↠   The breaking of laminar boundary layer and make it turbulent boundary layer by making the surface of the plate rough and or installing tripping wire.

Boundary Layer Thickness

The Prandtl/Blasius Boundary Layer Solution

• Viscous incompressible flow past any object can be obtained by solving the Navier-stokes equation.
• For steady, 2D laminar flows with negligible gravitational effects, Navier-stokes and Continuity equation reduce to:

Assumptions for simplifications are:

1. Since the boundary layer is thin  ↠  perpendicular velocity v << stream-wise velocity u.
2. Zero pressure gradient (potential flow and constant velocity).
3. Within the boundary layer, the viscous force is comparable to the inertial forces.

So above equations with the application of boundary conditions ( u = v = 0  at  y = 0, u ↠ U  as  y ↠ ∞), we get:

Blasius Solution

• Only valid for Laminar Flow.
• The boundary layer thickness grows as, "the square root of displacement 'x' and inversely proportional to the square root of velocity 'U'".

• The Blasius equation is a 3rd order Differential equation which can be solved by Standard methods (Runge Kutta). No analytic solution has yet been found.

References:

• Material from Class Lectures + Book named Fundamentals of Fluid Mechanics by Munson, Young & Okiishi's (8th Edition) + my knowledge. 
• Pics and GIF from Google Images.  
• Videos from YouTube (Engineering Sights).