Classification of Fluid Flow Regimes

The wind tunnel can be defined as a system that can be used to produce a rapid air stream through a section of testing in which a series of objects or an object is located. The major reasons for carrying out such tests are to evaluate the structural and/or aerodynamic effect of the object under consideration through simulating travel through wind or air on a structure. Wind tunnels are normally used in testing buildings, bridge structures, ground vehicles, wind sections, and aircraft, all in small scale.

Classification of Fluid Flow Regimes

Math number (M) can be defined as a quantity with no dimension representing the ratio of the speed of sound to the flow velocity past a boundary.

Math Number, M =, where c is the speed of sound in the medium and u is the local velocity relative to the boundaries. Math number, M can range from 0 to ?, however, this broad range can be grouped into numerous flow regimes. These regimes include hypersonic flow, supersonic flow, sonic flow, transonic flow, and subsonic flow.

Hypersonic flow: This is when the flow math number is greater than 5 as per the thumb rule (Childress, 2009).Supersonic flow: This is when the flow Math number is greater than everywhere in the domain. This flow is not pre-warned because the speed of the fluid is greater than the speed of sound.

Sonic flow: This is when the flow Math number is equivalent to one.Transonic flow: This is when the flow Math number is between the range 0.8 and 1.2. Highly unstable and mixed supersonic and subsonic flows are the major characteristics of this regime (Gülçat, 2015).Subsonic flow: This is when the flow Math number is in the range of 0 to 0.8 or below 1, i.e., the fluid velocity is lower than the acoustic speed. However, flow Math number varies while passing through a duct or over an object. 

Drag, Lift, Thrust, and Pressure

Pressure

Pressure is the amount of force applied perpendicularly to the object surface per unit area. Mathematically, pressure, P can be expressed as:P =; Where F is the magnitude of the normal force A is the contact surface area. The most important variables when calculating pressure are the Force and Area of the surface in contact (Jakobsen, 2014).

Thrust

Thrust can be defined as the force applied to a surface in a direction normal or perpendicular to the surface. Thrust, T can be mathematically expressed as:

Drag, Lift, Thrust, and Pressure

T = v   where v denotes the speed of the exhaust measured relative to the rocket and  is mass flow rate or the rate of variation of mass relative to time. The most important variables when calculating thrust are the velocity, mass, and time of surface in contact.

Lift

Lift is a force that conventionally acts in an upward direction so as to counter the gravitational force, however, the lift force can also act in a given direction at a right angle to the flow. In the case of an aeroplane, lift, L can be calculated as:

L = ½ ?v²SC_L

Where; C_L is the lift coefficient, S is the wing area, v is the velocity, ? is the air density, and L is the lift force. The most important variables when calculating lift are the area of the surface, the velocity of the fluid, fluid density, and lift coefficient (Ledoux, 2017).

Drag

This is a force acting opposite with respect to the relative motion of an object relative to the surrounding fluid. Drag force can be calculated as:

F_D = ½ ?v²C_DA; Where A is the area of cross-section, C_D is the drag coefficient, v is the velocity of the object with respect to the fluid, ? is the fluid density, and F_D is the drag force. The most important variables when calculating drag are the area of the surface, the velocity of fluid, fluid density, and drag coefficient (Palmer, 2011).

Bernoulli’s Principle in the Theory of Lift Generation

The generation of lift is normally explained in terms of Bernoulli’s Principle which states that for an incompressible, steady, and non-viscous fluid, the sum of kinetic, potential energies, and pressure per unit volume at any point is constant. When potential energy is ignored because of altitude, the Bernoulli’s Principles states that when the velocity of a fluid increases, there is a decrease in pressure by an equal quantity to maintain the overall energy (Pillai, 2010).

From the Bernoulli’s Principle, the passing air over the wing’s top or aerofoil must propagate faster and further compare to a similar air propagating the shorter distance below the wind in the same moment but the energy related with air must be constant at every moment. The results of this is that the above air the aerofoil has lesser pressure compared to the air beneath the wing and this variation in pressure results in the generation of lift (Rathore, 2010).

Bernoulli’s Principle in the Theory of Lift Generation

The principle of Velocity Measurement by Pitot tube

The pitot tube is an adaptable and reliable measuring device for fluid velocity. The pitot tube is used in airspeed sensor in the nose or wing of an aircraft, but the device can be used to measure the velocity of any fluid moving.

Figure 1: Pitot tube (Vennard, 2013)

The pitot tube works by the measuring the difference between pressure due to the momentum or velocity and the static pressure of the flowing fluid of the fluid molecules. The static pressure, P_stat, can be defined as the pressure without considering the motion and is determined by inserting an open-tube probe into the fluid so that the opening is parallel to the flow direction. The total pressure (stagnation) which is the pressure due to fluid momentum can be determined by inserting an open-tube probe facing upstream directly. In case both the probes are joined to two sections of diverse pressure gauge as shown below, the gauge reading is the pressure difference and is known as velocity head (Vennard, 2013).

ΔP = velocity head = P_tot – P_stat …………………………………………….. (i)

Work per unit mass can be determined by:

W = (P_tot – P_stat)/ ? ………………………………………………………….. (ii)

Since the only form of storable energy that varies in equation (i) is kinetic energy, then:

(P_tot – P_stat)/ P = -(ke) = (V_s²)/2 – (V_tot²)/2 …………………………………… (iii)

Where ½ v² is the kinetic energy per unit mass and Vs is the velocity of the undisturbed flowing gas. V_tot = 0 since it is the velocity at the stagnation point. Hence, the velocity of moving fluid is:

Vs = {2 * ((P_tot – P_stat)/ ?)^1/2 or ………………………………………………… (iv)

Vs = {2 * ΔP_p/}^1/2  …………………………………………………………….. (v) 

Magnus Effect

Magnus effect or force is observed when a rotating sphere in uniform flow experiences a lift which causes the object to drift across the flow direction. The rotation of a sphere that is rigid will result in the surrounding fluid to be entrained. When the sphere is positioned in a uniform flow, this will result in higher angular velocity on one side of the sphere, and lower angular velocity on the other section of the sphere. This provides an asymmetrical pressure distribution around the object (Wight, 2012). This will result in a lift on the sphere that moves the sphere towards the area of higher local angular velocity as shown in the figure below:

Figure 3: Magnus lift force on a rotating sphere (Wight, 2012)

Magnus force, FM can be expressed as:

F_M = ½ C_L?v²A; Where C_L is the coefficient of lift, A is the characteristic area, v is velocity magnitude, and ? is fluid density

Conclusion

A wind tunnel is also an important tool in education for studying aerodynamics and fluid dynamics. The investigation of the aerodynamic behaviour of a product can be used in improving the efficiency and prevention of disastrous failure. Such as experiments of wind tunnels on cars can minimize consumption of fuel by reducing drag or aerodynamics of bridge can be performed so as to evaluate the wind forces that will be exerted on it. 

Reference

Childress, S., 2009. An Introduction to Theoretical Fluid Mechanics. London: American Mathematical Soc.

Gülçat, Ü., 2015. Fundamentals of Modern Unsteady Aerodynamics. Michigan: Springer.

Jakobsen, H., 2014. Chemical Reactor Modeling: Multiphase Reactive Flows. London: Springer Science & Business Media.

Ledoux, M., 2017. Fluid Mechanics. Colorado: John Wiley & Sons.

Palmer, G., 2011. Physics for Game Programmers. Sydney: Apress.

Pillai, N., 2010. Principles Of Fluid Mechanics And Fluid Machines. Berlin: Universities Press.

Rathore, M., 2010. Thermal Engineering. New York: Tata McGraw-Hill Education.

Vennard, J., 2013. Elementary Fluid Mechanics. Melbourne: Read Books Limited.

Wight, G., 2012. Fundamentals of Air Sampling. California: CRC Press.

Zohuri, B., 2017. Thermal-Hydraulic Analysis of Nuclear Reactors. Perth: Springer.