NORMAL SHOCK WAVE

CONTENTS1. Introduction…………………………………………………………………………….…42. Thermodynamic Properties…………………………………………………………....63. Types of Thermodynamic Processes………………………………………………....124. Speed of Sound………………………………………………………………………….165. Mach Number……………………………………………………………………………226. When is a Flow

Compressible?.............................................................................237. Shock Wave………………………………………………………………………………278. Normal Shock Wave…………………………………………………………………….299. Governing Equations for Normal Shock Wave…………………………………….3210. Special Forms of Energy Equations…………………………………………………..35

11. Calculation of Normal Shock Wave Properties………………………….38 12. Measurement of Velocity in a Compressible Flow……………………...45

INTRODUCTIONFluid is a substance that deforms continuously under the action of tangential stresses. Fluid

mechanics is the branch of applied mechanics in which the behaviour of fluid is studied under the action of forces. It is classified into three categories namely as fluid statics, fluid kinematics and fluid dynamics. If the fluid is at rest, it’s study is said to be fluid statics. If the fluid is in motion, it is known as fluid kinematics. In the above mentioned types, external forces (body and surface forces) are not considered. If these forces are considered, then fluid is said to be fluid dynamics. Fluid dynamics is further classified into two categories, namely as hydrodynamics and aerodynamics. In hydrodynamics, effects of water are studied in different channels, while in aerodynamics, effects of air studied over the solid bodies.

Aerodynamics is an applied science with many practical application in engineering. The flow of air over objects is of fundamental interest to aeronautical and space engineers in the design of air crafts, missiles and rockets. The knowledge of fluid mechanics is used to maximize lift and minimize drag on these bodies. The engineers of aerodynamics are always interested to increase the velocity of objects (jets, rockets, etc.).

Fluid may also be classified into two categories, namely, compressible and incompressible fluid. The fluid with variable density is known as compressible, however, if the density remains constant, it is said to be incompressible fluid. Mostly, compressible flows are discussed in aerodynamics.

Flows are further categorized with the help of Mach number. Flow is subsonic if, flow is sonic if, flow is supersonic if and hypersonic if. The recent research, in this field, is mostly concentrated about supersonic and hypersonic flows. The significant topic for research, in these flows, is the shock waves. It usually appear in supersonic flows and is just like a boundary layer of thickness about cm. The topic of our concentration in this work is the study of shock waves in supersonic flow. The shock waves are classified as normal shock waves and oblique shock waves. A normal shock wave is created by a blunt body in supersonic flows while an oblique shock wave is a sharp edge shock wave that is formed when supersonic flow is turned on itself. Here we will only discuss about the normal shock wave, it’s governing equations and the calculation of it’s properties. Finally, we shall conclude that how the velocity of a normal shock wave can be determined from these calculations.

We have studied the flows for which all quantities (velocity, pressure, density, etc.) change continuously. Flows are also possible for which these quantities vary discontinuously. Such discontinuities occur in explosions, detonations, supersonic movements, powerful electric discharge and other phenomena’s that create an extreme change in pressure. These discontinuities occur when large pressure change in a small volume is produced, the resulting pressure wave moves at a velocity much higher than the velocity of sound. This pressure wave is known a shock wave that causes an abrupt change in pressure, density, temperature, etc.

As there occurs extreme changes in thermodynamic properties (temperature, pressure, etc.) of the fluid. So, before starting our brief discussion on the normal shove wave, Let us take a little glance on these properties. After this discussion, we will also take the knowledge about the speed of sound.

THERMODYNAMIC PROPERTIESThermodynamics is defined as study of the relationship between the heat and

other forms of energy. It plays an important role in the flow of gas, particularly for high speed flow i.e. flow around high speed flight of aircrafts and missiles.

Ideal Gas A gas in which intermolecular forces are neglected is known as an ideal gas. It is also known as a perfect gas. The equation of an ideal gas is given as

Where is the specific gas constant. It has different values for different gases.

It is sometime convenient to define the values of thermodynamic properties per unit mass; such values are distinguished by the word “specific”.

Specific Volume

It is defined as the volume occupied by a unit mass of the fluid

and is denoted by

Specific WeightIt is defined as the weight (force due gravity) per unit volume and is defined as

Where is the weight per unit volume.

Specific Internal Energy

It defined as the energy per unit mass of the fluid due to molecular activity and is denoted by.

Specific EnthalpyIt is defined as the total heat introduced per unit mass of the fluid and is denoted

by

Specific HeatIt is defined as the amount of heat required to raise the temperature of a unit

mass of fluid by one degree and is denoted by

Specific Heat At Constant VolumeIt is denoted and defined as

Where is the specific internal energy of the gas.Specific Heat At Constant Pressure

It is denoted and defined as

Where is the specific enthalpy of the gas.

Ratio of Specific HeatsIt is defined as a measure of the relative internal complexity of the molecules of

the fluid and is denoted by

For air, .

TYPES OF THERMODYNAMIC PROCESSESWhen gases are expanded or compressed, the relationship between the pressure, temperature, and density depends on the nature of the process. The types of thermodynamic processes are discussed below:

Reversible ProcessThe process in which no dissipative phenomena occur i.e. the process occurs in such

a manner that it can be returned to its original state.

Irreversible processThe process that is not reversible is known as irreversible process.

Isobaric ProcessIt is defined as the process in which the pressure of a gas is kept constant during

its expansion or compression.

Isothermal ProcessIt is defined as the process in which the temperature is kept of a gas constant

during its expansion or compression

Isometric ProcessIt is defined as the process in which the volume is kept of a gas constant.

Adiabatic ProcessA process in which no heat is added to or taken away from a gas during expansion

or compression. In this case

Isentropic ProcessA process which is reversible and adiabatic is said to be Isentropic Process.

First Law of ThermodynamicsIt states that the heat added and work done on gas cause a change in energy of a

gas. Its mathematical form is

Second Law of ThermodynamicsIt states that the entropy change is greater than the heat transferred to the

system divided by the temperature.

For an adiabatic process, , then

SPEED OF SOUNDCommon experience shows that sound travel through air at some finite velocity.

The physical mechanism of sound propagation in a gas is based on the molecular motion. For instance, imagine that you are sitting in a room and suppose that a firecracker detonates in a corner. When it detonates, chemical energy (in the form of heat) is transferred to the air molecules adjacent to the firecracker. These energized molecules move about in a random fashion. They ultimately collide with their neighbouring molecules and transfer their energy in the process. In turn, these molecules collide with their neighbours and transfer their energy. In this fashion, the energy released by the firecracker is transferred through the air by molecular collisions. Finally, as this energy wave from the cracker passes over our eardrums, we hear the slight pressure changes in the wave. This is sound, and the propagation of the energy wave is actually the propagation of a sound wave through the gas.

Now, let us derive an equation for the speed of sound in a gas. In our above discussion, note that the propagation of sound wave is due to the molecular collisions of the gas. This is the microscopic view of our analysis and we will not use it for the derivation of the result. We take benefit of the fact that the macroscopic properties changes across the wave and use our macroscopic equations of continuity, momentum and energy to analyse these changes.

Consider a sound wave propagating through a gas with velocity, where the local pressure, temperature and density are , as shown in figure 3. These are the gas properties in front of the sound wave. Behind the wave, the gas properties pressure, temperature and density are slightly different and given by, respectively. The changes are very small-infinitesimal. Assume that flow is one dimensional, and isentropic (i.e. adiabatic and reversible). The flow can be made steady by considering an observer moving along with the sound wave. For such an observer, it seem to be stationary. Let us apply the

governing equations to obtain our result.

Applying the continuity equation (conservation of mass) for the flow, we have

The L.H.S of the above equation is the mass inflow while the R.H.S shows the mass outflow. The product can be neglected because and are infinitesimal. Hence, solving above equation for, we have

Now consider the momentum equation (conservation of momentum) for the above mentioned flow, we have

=

Ignoring the products of differentials, the above equation becomes

Solving equation for we have

Solving equations

Rewrite the above equation, we have

The above equation is the fundamental equation for the speed of sound in a gas. Assume that gas is calorically perfect, then the isentropic relation for such a gas is given as

Differentiating above equation and then solving for, we obtain

The above result is the equation for the speed of sound in a calorically perfect gas. From the equation of an ideal gas, , the above equation becomes

Which is the final expression for the speed of sound: it states that the speed of sound in a calorically perfect gas is a function of temperature only. The value of speed of sound at standard sea level is

MACH NUMBERIt is a dimensionless number and is defined as the ratio of the velocity of fluid to

the local velocity of sound in the same medium. It is used to measure the compressibility in a fluid. It is denoted by with the mathematical expression

WHEN IS A FLOW COMPRESSIBLEConsider a fluid particle of gas initially at rest with density . Assume flow is

isentropic and moves with velocity and Mach number . As the velocity of fluid particle increases, the other flow properties will change according to the basic governing equations. The density of the fluid particle will change according to the following equation

The above equation states that only the value of dictates the ratio .

For an isentropic process, the ratios for total to static pressure and density are given as

For γ = 1.4, the variation of is plotted as a function of from zero to sonic flow, shown in the figure 4. Examine that at low subsonic Mach numbers, the variation is relatively very flat. In fact, for the value of deviates from by less than 5 percent, and for all practical purposes the flow can be treated as incompressible. However, for , the variation in is larger than 5 percent, and its variation becomes more pronounced as increases. As a result, many aerodynamicists have adopted the rule of thumb that density variation should be accounted for at Mach numbers above : i.e. flow should be treated as compressible.

5 % variation

For ,variation in is less than 5 %

M

Figure 4: isentropic variation of density with Mach number

In the above discussion, we have talked about some aspects of thermodynamic properties, and speed of sound as well. The condition for which the fluid becomes compressible, is also come under discussion. We are now in situation to start a brief discussion about a normal shock wave.

SHOCK WAVEAs discussed earlier, a shock wave is classified into two categories namely as

oblique shock wave and normal shock wave. If the shock wave makes a certain angle in the direction of flow stream, it is said to be an oblique shock wave. It occurs over a wedge shaped bodies or a concave corner, as shown in figure 5. The creation of normal shock wave will be discussed briefly in our next discussion.

Figure 5: oblique shock wave over a wedge.

NORMAL SHOCK WAVEIf a shock wave occurs normal to the direction of flow, it is said to be normal shock

wave. Note that it is the special case of oblique shock wave i.e. if a shock wave makes an angle of 90° with the flow direction, it becomes a normal shock wave. It may occur in any supersonic flow. For example, it may occur through constant area nozzle or a diverging duct, or in front of a blunt-nosed body. Two examples are given in this regard. In the first one, shown in figure 1, Supersonic flow is established through a nozzle which can be a supersonic wind tunnel.

Figure 1: Normal shock through nozzle.

In the second example, shown in figure 2, a supersonic flow over a blunt-nosed body which can be the front face of a fighter jet. A strong shock wave exists in front of the body. Although the shock wave is curved but the region of the shock wave closest to the nose is essentially normal to the flow direction.

Figure 2: Normal shock wave in a blunt body.

There are many other examples in which this phenomena may occur. In the next section, we will obtain the basic normal shock equations.

GOVERNING EQUATIONS FOR NORMAL SHOCK WAVE

Consider the normal shock wave shown in figure 3. Assume that flow is steady, one dimensional and adiabatic with no body forces. The area is assumed to be constant throughout the normal shock wave. The region in front of the upstream shock is a uniform flow, and the region behind the downstream shock is a different uniform flow. The velocity, temperature, pressure, density, Mach number, total pressure, total temperature, total enthalpy, and entropy in front of the shock are and . The corresponding quantities behind the shock are and.

The continuity equation for an inviscid, compressible flow under the above assumptions is obtained as

The above equation is the continuity equation for normal shock waves. It states that volume flow across the shock wave is constant.

The momentum equation for an inviscid, compressible flow under the above assumptions is obtained as

Which is the momentum equation for normal shock waves.

Now the energy equation for the above mentioned flow is given by

Which is the energy equation for normal shock waves. It states that enthalpy is constant across the normal shock wave.

Note the above equations closely. The quantities upstream of the shock wave, etc. are known. We have a system of three algebraic equations with four unknowns and. If we add the following thermodynamics relations

And

Now we have five equations with five unknowns, namely, and. The above equations can explicitly be solved to obtain the values of unknowns.

SPECIAL FORMS OF ENERGY EQUATIONThe energy equation for an inviscid, steady, one dimensional and adiabatic flow

is given as

= +

For a calorically perfect gas, the ratio of total temperature to static temperature is a function of Mach number only. It’s relation is given as

The above equation states that only the value of dictates the ratio .

For an isentropic process, the ratios for total to static pressure and density are given as

And

Note that the above ratios are the functions of . From these ratios, the quantities can be calculated from the actual conditions.

Consider the case when flow is exactly sonic i.e. . The static temperature, pressure, and density, at sonic condition, are denoted by and . Then the above relations become

From the definition of Mach number, , where is the speed of sound. Let us introduce a “characteristic” Mach number defined as

Where is the speed of sound at sonic condition. The relationship between the actual Mach number and the characteristic Mach number is given as

CALCULATION OF NORMAL SHOCK WAVE PROPERTIES

Let us move to drive the relations which are helpful in the calculation of changes of flow properties across a normal shock wave. We know the relations of continuity, momentum, and energy for an inviscid, steady, one dimensional, and isentropic flow

Continuity:

Momentum:

Energy:

For a calorically perfect gas, we know that

And

These are five equations with five unknowns, namely, and. Hence, these equations are sufficient to find the properties behind a normal shock wave in a calorically perfect gas. Let us proceed to obtain different relations.

First, solving equations (), () and (), after doing some algebraic calculations, we obtain

The relation is known as the Prandtl relation.

By the definition of characteristic Mach number, , we have from Prandtl relation

Using the value of and after some algebraic simplification, we get a result

The above result is first major result for a normal shock wave. It states that is the only function of . We now derive the relations for the ratios of the thermodynamics properties , , and across a normal shock wave.

To obtain the density ratio, solve equations () and (), then using the value of ,

We get

To obtain the pressure ratio, solve equations () and (), after simplification, we obtain

To obtain the temperature ratio, we proceed as follow:

From the equation of state , we have

Using equations () and () in the above relation, and from the equation ,

We get

From the above results, note that the Mach number is the determining parameter for changes across a normal shock wave in a calorically perfect gas.

As discussed earlier, shock wave occurs in supersonic flows () i.e. it does not occur in subsonic flows (). Note that in equations (), (), (), and (), . However, these equations are also solvable for (on mathematical basis). This issue can be resolved with the help of second law of thermodynamics. Apply the second law to the flow across a normal shock wave.

The relation of the entropy change across a normal shock wave is given as

Using equations () and (), we get

Note that entropy change across the shock is a function of only. The second law states that

In the above equation, if we have

.

If , we have

Both of the above results obey second law of thermodynamics. If , we have

Which disobey the second law. Hence, in nature, normal shock wave occurs only in supersonic flows ().

MEASUREMENT OF VELOCITY IN A COMPRESSIBLE FLOW

The velocity of a low speed, incompressible flow can be measured by pitot tube. In this measurement, the total pressure is measured by pitot tube, and the static pressure is measured from a static pressure orifice. In this section, we assume compressible flow, both subsonic and supersonic. Here we will consider Mach number rather than the velocity. Let us proceed:

Subsonic Compressible FlowConsider a subsonic, compressible flow. A pitot tube is inserted in this flow to measure the Mach value. The total pressure is of the freestream is denoted by and the static pressure of the freestream is denoted by , then Mach number can be obtained from the equation

Or

In the above equation, total pressure and static pressure help to find the value of Mach number.

SUPERSONIC COMPRESSIBLE FLOW

Consider a pitot tube is inserted in a compressible, supersonic flow. Because of supersonic flow, there will be a strong bow shock wave in front of the tube as shown in figure. In this case, the total pressure is behind the normal shock wave, . The Mach value can be obtained by using the total pressure and static pressure as follows:

Where is the ratio of the total to static pressure behind the shock and is the static pressure ratio across the shock.

From equation (), (), and ()

and

And

Solving above three equation, we obtain

The equation is known as the Rayleigh Pitot tube formula. It gives the Mach value for the known value of .

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