Model Study /Fluid Mechanics

Modeling

An engineering model is a physical or mathematical representation of a prototype engineering system design that is utilized to predict some aspects of the prototype behavior. Physical models of smaller size may be used in wind tunnels to obtain information on the prototype flow around airplanes, automobiles and skyscrapers. With the advent of supercomputers, flow over aircraft under conditions that cannot be simulated in a wind tunnel can now be calculated using mathematical models of such flow. Short range forecasts of weather conditions are aided by the exercising of mathematical models of the atmospheric circulation system. In this section, we will concentrate on the physical modeling of engineering systems as carried out in laboratory experiments.

The use of engineering models goes back at least to 1872 when William Froude built the first towing tank to test the resistance of ocean vessels. A small wind tunnel was utilized by the Wright brothers to develop their first aircraft. Nowadays wind tunnels, towing tanks and hydraulic models of dams, rivers and harbors are commonplace aids to the design of engineering systems. The usefulness of such models relies

upon the principles that underlie the relation, called similitude, between the model characteristics and those of the prototype.

Similitude

In geometry we learned how to prove the similarity of plane figures, such as triangles and rectangles. Similar figures had the same shape but different sizes. Corresponding angles must be equal and the ratio of the lengths of corresponding sides also must be equal. This illustrates the principle of geometric similitude. As applied to modeling, it requires that the shapes of the model and the prototype be identical, but the size may be different by a geometric scale ratio SR that equals the ratio of their characteristic lengths:
 
where and are the same characteristic length (such as ship length or aircraft wing span) of the model and prototype, respectively.
If we were to take photographs of a model and its prototype, the photographs would be indistinguishable if taken at distances from the object that are in the same proportion to and .
If we are modeling an engineering structure that moves through a fluid, we must ensure that the model moves with time in a manner that is similar to that of the prototype. This will be achieved if the ratio of the acceleration of the structure to the acceleration of a fluid particle is the same for both model and prototype. For example, if we want to model a wind turbine of radius L rotating at an angular speed in a wind of speed , we must require that the ratio of the centrifugal acceleration of the turbine, , to a typical acceleration of a fluid element, , should be the same for both model and prototype:
 
which is equivalent to the equality of the Strouhal number for both model and prototype. Note that, in combination with equation 10.34, this implies a constraint on the value of the flow length :
 
This illustrates the principle of kinematic similitude, where the acceleration of the fluid and its boundaries are maintained in the same ratio for both model and prototype. Using the photographic analogy, kinematic similarity between model and prototype implies that moving pictures of the model and prototype flows would appear identical if the framing rates are proportional to and and the flow speeds satisfy the relation of equation 10.36.
Although geometric and kinematic similitude are necessary conditions for a model experiment to portray a prototype flow, they are not sufficient. We need to ensure that the entire flow fields of the model and prototype flows are similar. One way to assure ourselves that this will be so is to require that the solutions to the dimensionless conservation laws, as described in section 10.2.4, are identical for both model and prototype flows. For this to be true, the dimensionless variables such as etc., must be the same for both model and prototype since they appear as parameters in the dimensionless expression of the conservation laws. However, these dimensionless variables are the same as those 's that we derived in undertaking a dimensional analysis of a flow problem. We can then ensure a complete dynamical similitude between model and prototype flow by requiring that the model and prototype 's be equal:
 
The similarity of model and prototype flow fields is thus ensured if the length scales of all dimensions conform to geometric similarity (equation 10.34) and if the independent dimensionless 's are the same for both model and prototype (equation 10.37). These conditions enable us to relate the dimensional model and prototype variables to each other, thereby providing a prediction of the prototype variables from a measurement of the model values. This is the objective of the model experiments.
As an example of modeling, suppose that we construct a laboratory scale model of a complex piping system in order to investigate the pressure drops and flow rates in the system. To ensure similitude, we need a geometrically similar model of scale ratio SR (all pipe diameters, lengths and roughness heights are in proportion to SR) and all independent 's must be equal for both model and prototype. Using the dimensional analysis of equation 10.17 for pipe flow, the independent 's may be chosen to be the Reynolds number and the roughness ratio . Because we have a scale model, the roughness ratio equality is ensured:

by virtue of equation 10.34. The equality of the Reynolds numbers requires:
 
If we decide to use the same fluid in the model as is used in the prototype flow (although this is not necessary ), the model flow speed must be higher than the prototype speed by the factor .
Changes in pressure between any two points in the system will be proportional to the pressure gradient times the diameter D, but because the roughness ratio and Reynolds number of the model and prototype are each equal, so must be the friction factors (see equation 10.17):
 
where we have used equation 10.38 to eliminate the velocities in 10.39. If the fluids are the same for model and prototype, then the prototype differences in are smaller than the model values by the factor , for corresponding points in the piping system.

Applications of Modeling

Modeling of engineering fluid flows is commonly used in the design of aircraft, ships, road vehicles, and civil works, but there are many special cases of modeling that are useful in research and development, such as blood flow in elastic vessels or the flow of a buoyant plume from a smoke stack. In this section, we consider several examples of modeling flows in engineering systems.

Fluid Mechanics Assignment No.1

Fluid Mechanics

Assignment no. 1

1.      Is surface tension & capillarity dependent on density of liquid? If yes, How? If no, Why?

2.      On what factor does the capillarity of a liquid depends and how? Explain.

3.      Can a liquid be boiled at ordinary temp? (Room temp.). Justify.

4.      Explain Cavitation.

5.      How the Stability of Floating bodies effected by the position of centre of gravity & centre of buoyancy?

6.      Why the shop floats in water while the needle sinks? Explain.

7.      A Circular thin plate, 500mm dia. is immersed in water vertically such that its top edge is 2m below the free water surface. Find the total hydrostatic pressure acting on the plate & position of its centre of pressure.

8.      A triangle surface 1m base and 1.5m altitude is placed in a liquid of specific gravity 0.80. The plane of the area is inclined 30º to the liquid surface and the base is parallel to and at a depth of 2m from the free surface. Determine the total pressure and the position of centre of pressure.

9.      A triangle lamina of base 500mm and height 750mm is hanged vertically inside tank containing glycerin of specific gravity 1.26 such that its base coincides with the free surface. Determine the glycerin pressure acting on one side of the lamina and depth of centre of this pressure.

10.  a tank contains water upto a height of 1 m above the base a liquid or specific gravity 0.8 is filled on the top of water upto 1.5 m height calculate (i) total pressure on one side of the tank (ii) the position of centre of pressure for one side of the tank which is 3 m wide.

Submission  Date:- 31/1/12

Assignment No 1

Fluid Mechanics

Date of issue …………………...

Date of submission……………..

1.      Is surface tension & capillarity dependent on density of liquid? If yes, How? If no, Why?

2.      On what factor does the capillarity of a liquid depends and how? Explain.

3.      Can a liquid be boiled at ordinary temp? (Room temp.). Justify.

4.      Explain Cavitations.

5.      How the Stability of Floating bodies effected by the position of centre of gravity & centre of buoyancy?

6.      Why the shop floats in water while the needle sinks? Explain.

7.      A Circular thin plate, 500mm dia. is immersed in water vertically such that its top edge is 2m below the free water surface. Find the total hydrostatic pressure acting on the plate & position of its centre of pressure.

8.      A triangle surface 1m base and 1.5m altitude is placed in a liquid of specific gravity 0.80. The plane of the area is inclined 30º to the liquid surface and the base is parallel to and at a depth of 2m from the free surface. Determine the total pressure and the position of centre of pressure.

9.      A triangle lamina of base 500mm and height 750mm is hanged vertically inside tank containing glycerin of specific gravity 1.26 such that its base coincides with the free surface. Determine the glycerin pressure acting on one side of the lamina and depth of centre of this pressure.

10.  a tank contains water upto a height of 1 m above the base a liquid or specific gravity 0.8 is filled on the top of water upto 1.5 m height calculate (i) total pressure on one side of the tank (ii) the position of centre of pressure for one side of the tank which is 3 m wide.

stability----Fluid Mechanics

ARCHIMEDES   PRINCIPLE

The buoyant force on a submerged body

• The Archimedes principle states that the buoyant force on a submerged body is equal to the weight of liquid displaced by the body, and acts vertically upward through the centroid of the displaced volume.
• Thus the net weight of the submerged body, (the net vertical downward force experienced by it) is reduced from its actual weight by an amount that equals the buoyant force.

The buoyant force on a partially immersed body

• According to Archimedes principle, the buoyant force of a partially immersed body is equal to the weight of the displaced liquid.
• Therefore the buoyant force depends upon the density of the fluid and the submerged volume of the body.
•  For a floating body in static equilibrium and in the absence of any other external force, the buoyant force must balance the weight of the body.

Stability of Unconstrained Submerged Bodies in Fluid

• The equilibrium of a body submerged in a liquid requires that the weight of the body acting through its cetre of gravity should be colinear with an equal hydrostatic lift acting through the centre of buoyancy.
•  In general, if the body is not homogeneous in its distribution of mass over the entire volume, the location of centre of gravity G does not coincide with the centre of volume, i.e., the centre of buoyancy B.
• Depending upon the relative locations of G and B, a floating or submerged body attains three different states of equilibrium-

Let us suppose that a body is given a small angular displacement and then released. Then it will be said to be in

• Stable Equilibrium: If the body  returns to its original position by retaining the originally vertical axis as vertical.
• Unstable Equilibrium: If the body does not return to its original position but moves further from it.
• Neutral Equilibrium: If the body  neither returns to its original position nor increases its displacement further, it will simply adopt its new position.

Stable Equilibrium

Consider a submerged body in equilibrium whose centre of gravity is located below the centre of buoyancy (Fig. 5.5a). If the body is tilted slightly in any direction, the buoyant force and the weight always produce a restoring couple trying to return the body to its original position (Fig. 5.5b, 5.5c).

Fig 5.5    A Submerged body in Stable Equilibrium

Unstable Equilibrium

On the other hand, if point G is above point B (Fig. 5.6a), any disturbance from the equilibrium position will create a destroying couple which will turn the body away from its original position (5.6b, 5.6c).

Fig 5.6    A Submerged body in Unstable Equilibrium

Neutral Equilibrium

When the centre of gravity G and centre of buoyancy B coincides, the body will always assume the same position in which it is placed (Fig 5.7) and hence it is in neutral equilibrium.

Fig 5.7    A Submerged body in Neutral Equilibrium

Therefore, it can be concluded that a submerged body will be in stable, unstable or neutral equilibrium if its centre of gravity is below, above or coincident with the center of buoyancy respectively (Fig. 5.8).

Fig 5.8   States of Equilibrium of a Submerged Body

(a) STABLE EQUILIBRIUM    (B) UNSTABLE EQUILIBRIUM      (C) NEUTRAL EQUILIBRIUM

 Stability of Floating Bodies in Fluid

• When the body undergoes an angular displacement about a horizontal axis, the shape of the immersed volume changes and so the centre of buoyancy moves relative to the body.
•  As a result of above observation stable equlibrium can be achieved, under certain condition, even when G is above B.
  Figure 5.9a illustrates a floating body -a boat, for example, in its equilibrium position.

Fig 5.9     A Floating body in Stable equilibrium

Important points to note here are

1. The force of buoyancy F_B is equal to the weight of the body W
2. Centre of gravity G is above the centre of buoyancy in the same vertical line.
3. Figure 5.9b shows the situation after the body has undergone a small angular displacement q with respect to the vertical axis.
4. The centre of gravity G remains unchanged relative to the body (This is not always true for ships where some of the cargo may shift during an angular displacement).
5. During the movement, the volume immersed on the right hand side increases while that on the left hand side decreases. Therefore the centre of buoyancy moves towards the right to its new position B'.

Let the new line of action of the buoyant force (which is always vertical) through B' intersects the axis BG (the old vertical line containing the centre of gravity G and the old centre of buoyancy B) at M. For small values of q the point M is practically constant in position and is known as metacentre. For the body shown in Fig. 5.9, M is above G, and the couple acting on the body in its displaced position is a restoring couple which tends to turn the body to its original position. If M were below G, the couple would be an overturning couple and the original equilibrium would have been unstable. When M coincides with G, the body will assume its new position without any further movement and thus will be in neutral equilibrium. Therefore, for a floating body, the stability is determined not simply by the relative position of B and G, rather by the relative position of M and G. The distance of metacentre above G along the line BG is known as metacentric height GM which can be written as

GM = BM -BG

Hence the condition of stable equilibrium for a floating body can be expressed in terms of metacentric height as follows:

GM > 0 (M is above G)                                      Stable equilibrium
GM = 0 (M coinciding with G)                          Neutral equilibrium
GM < 0 (M is below G)                                      Unstable equilibrium

The angular displacement of a boat or ship about its longitudinal axis is known as 'rolling' while that about its transverse axis is known as "pitching".

Floating Bodies Containing Liquid

If a floating body carrying liquid with a free surface undergoes an angular displacement, the liquid will also move to keep its free surface horizontal. Thus not only does the centre of buoyancy B move, but also the centre of gravity G of the floating body and its contents move  in the same direction as the movement of B. Hence the stability of the body is reduced. For this reason, liquid which has to be carried in a  ship is put into a number of separate compartments so as to minimize its movement within the ship.