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The correct OPTION is (d) Thermal dissipation there

The best EXPLANATION: The flow across the normal shock wave is isentropic i.e. it is adiabatic also. Since, we are concerned with calorically PERFECT GASES in an adiabatic, INVISCID, steady flow the total temperature remains constant. Thermal dissipation is not a reason for total temperature being constant.

Right ANSWER is (b) True

The best I can explain: This comes from the physical meaning of the Mach NUMBER. When we take the ratios of per unit mass kinetic ENERGY to the potential energy of the fluid particle MOVING along a STREAMLINE, it comes proportional to the square of the Mach number. Thus, this is true.

Correct choice is (a) False

Easy explanation: The NORMAL shock wave, i.e. the ONE normal to the UPSTREAM flow, seems rare. But actually, it is a very frequent case of shock waves and occurs at many times in DIFFERENT cases.

Right choice is (b) SPEED of sound in incompressible medium is zero

To explain I would SAY: Incompressible flow is the limiting CASE of isentropic compressibility being zero. This gives an infinite speed of sound and a zero Mach NUMBER in that medium. Thus, theoretically, zero-Mach number flows are incompressible.

The correct option is (c) MACH NUMBER

Easy explanation: Unlike the incompressible flow, for compressible flow, even if it’s subsonic, we need other parameters to calculate speed of flow. With the KNOWLEDGE of static and stagnation pressure we can calculate Mach number directly, and hence it is not required to be SEPARATELY measured. We need free-stream velocity of SOUND to get flow velocity from Mach number.

Right choice is (b) False

To elaborate: When the SINGLE velocity is GIVEN, either upstream or downstream, a host of various temperatures will give a set of various Mach numbers, and hence different normal SHOCK waves. But if the TEMPERATURE is given along with the velocity, both either upstream or downstream, it specifies the normal shock wave.

The CORRECT choice is (c) Shock wave is pretty thick

The best I can explain: The shock waves are a very thin region and have entropy increase ACROSS them. The normal shock wave has LARGE velocity and temperature gradients across it and hence the irreversibility. The thermal conduction and frictional effect lead to increase in entropy.

Right option is (b) M1 > then M2 > 1

Easiest explanation: For a normal shock wave, the UPSTREAM and downstream MACH numbers are related irrespective of the other parameters for a particular FLOW. According to the relation, M1=1 then M2=1 andM1 > thenM2 < 1, since normal shock wave compresses the flow. Also, by Prandtl relation when M\(_1^*\)=1→M\(_2^*\)=1 and whenM1→∞ then M2 takes a finite value.

The correct option is (d) M→∞, M*→0

The best explanation: The characteristic Mach number and the Mach number behave ALMOST in a similar pattern. The only difference if when the Mach number tends to infinity, the characteristic Mach number tends to a finite value. This finite value DEPENDS only on GAMMA is NEVER zero.

The correct ANSWER is (b) γ , M

For EXPLANATION: The stagnation and static temperatures for a calorically perfect gas are related with the equation \(\FRAC {T_0}{T}\)=1+\(\frac {\gamma -1}{2}\)M^2. Thus, it is CLEARLY SEEN that this ratio depends on the gamma and the Mach number in the medium. It’s a very important relationship.

Correct option is (a) \(\frac {\GAMMA +1}{2(\gamma -1)}\) a^*2=\(\frac {a_0^2}{\gamma -1}\)≠const

The best explanation: The TWO properties, a* and a0, of the flow are RELATED to each other by the relation: \(\frac {\gamma +1}{2(\gamma -1)}\) a^*2=\(\frac {a_0^2}{\gamma -1}\). These two (a* and a0) are defined QUANTITIES and are constant at a point in the flow. Thus, the correct relation is \(\frac {\gamma +1}{2(\gamma -1)}\) a^*2=\(\frac {a_0^2}{\gamma -1}\)=const, since gamma is also a constant.

Right option is (B) ρ1u1=ρ2u2

The EXPLANATION: The continuity equation of the normal shock equation is DERIVED from the continuity equation of mass and is given as ρ1u1A1=ρ2u2A2. If the area of the control volume is same on both sides, it becomes ρ1u1=ρ2u2. The flow is steady and WITHOUT viscosity.

Correct answer is (d) STATIC pressure

Explanation: The velocity of SOUND can be calculated using the PITOT TUBE which measures the stagnation, also called total pressure and MEASURING the static pressure as well. There is no need to find temperature. And static velocity is nothing but zero velocity.

Right choice is (a) False

The explanation: Equations of mass, momentum, energy are valid in both super and subsonic mediums and HENCE shock wave can occur in both mediums. But, thermodynamics delete the possibility of SHOCKWAVES occurring in subsonic MEDIUM. Entropy change is negative if the upstream medium is subsonic and hence no shock WAVES occur in subsonic medium, else second law is violated.

The correct answer is (c) \(\FRAC {\RHO ^*}{\rho_0}\)=0.634

Best EXPLANATION: The defined flow properties are related to each other with relations involving only MACH number and gamma. Hence, when these TWO values are given, the ratios can be calculated. These are important ratios and for M=1 and gamma=1.4 the values are \(\frac {T^*}{T_0}\)=0.833, \(\frac {P^*}{P_0}\)=0.528, \(\frac {\rho ^*}{\rho_0}\)=0.634

Right option is (b) Free-STREAM static pressure

For explanation: The Rayleigh Pitot tube FORMULA HELPS to calculate the free- stream Mach number using the MEASURED values of free-stream static pressure and Pitot pressure. It is based on the fact that a bow shock is formed at the mouth of the Pitot tube for a COMPRESSIBLE supersonic flow. The Pitot pressure is the stagnation pressure behind the bow shock.

Right OPTION is (a) True

For explanation: Strictly speaking, all FLOWS are compressible i.e. incompressible flow is a myth ACTUALLY. But for all practical applications, flow with Mach number < 0.3 can be assumed incompressible SINCE the density VARIATION is less than 5%.

The correct ANSWER is (c) DIFFERENTIAL form of CONSERVATION equations used

The explanation: The normal shock wave analysis uses the control volume approach. The integral form of conservation equations are applied to the control volume. The flow is steady, adiabatic, without viscous effects and zero body FORCES.

Correct OPTION is (B) \(\FRAC {1}{p}\)

The explanation is: The isothermal compressibility differs from the isentropic compressibility by a factor of gamma. The isothermal compressibility is GIVEN as τt=\(\frac {1}{p}\) while the isentropic compressibility is given as τs=\(\frac {1}{\gamma p}\).

Correct option is (a) False

To ELABORATE: For the compressible flow- both supersonic and subsonic, we can CALCULATE the Mach number by using the Pitot tube. It measures the TOTAL pressure and by MEASURING the static pressure we can get the Mach number, not VELOCITY, unlike the incompressible flow.

The correct OPTION is (b) Isentropic COMPRESSION

Explanation: The defined properties of the flowP0 and ρ0 involve isentropic compression, which is adiabatic and reversible both. These properties are the VALUES of the flow parameters when the flow is isentropically compressed to zero VELOCITY, at a POINT with properties P and ρ.

The correct ANSWER is (a) ISENTROPIC

For EXPLANATION: The equation a=-ρ\(\FRAC {da}{d\rho }\) is the equation for the speed of sound in the medium. This is derived from the governing equations for sound WAVES, which are isentropic (adiabatic and reversible). Sound waves may or may not be compressible.

The CORRECT ANSWER is (d) a0 is the STAGNATION speed of sound

For explanation: For a point in the FLOW, where the speed of sound is a, a0 is the stagnation speed of sound associated with that point. While a* is the sonic characteristic value associated with that same point. None of these is the maximum speed of sound in the flow.

Correct answer is (B) Upstream Mach number

The BEST I can explain: Specifying a dimensionless quantity across the normal SHOCK wave will specify all the other ratios and both the upstream and downstream Mach NUMBERS. But for finding the velocity, temperature also needs to be specified and vice versa.

Right option is (B) GAS constant is same for both

The explanation: The sound of speed depends on GAMMA, T and the gas constant R for the respective gas. In case of helium and air, the gamma for helium is higher than air. Also, helium is lighter than air thus, R for helium being higher. This gives, at the same temperature, speed of sound more in helium than air.

The CORRECT ANSWER is (c) T only

Explanation: The speed of sound in any given medium is expressed generally using the equations INVOLVING pressure and density (P, ρ). But using the isentropic RELATIONS and the gas equation we can simplify it to show that speed of sound in any medium is a function of temperature only a=\(\SQRT {\gamma RT}\).

Right OPTION is (b) False

Explanation: The given statement is false. For a perfect gas, the SPEED of SOUND is a pressure of temperature only. And hence, by CHANGING pressure or DENSITY but keeping the temperature same, speed of sound will not change.

Correct CHOICE is (b) τs=\(\frac {1}{\rho a^2}\)

To EXPLAIN: The SPEED of sound in any MEDIUM is given by a^2=\(\frac {dp}{d\rho }\) and the isentropic compressibility is related to the speed of sound as τs=\(\frac {1}{\rho a^2}\). To derive this, v=\(\frac {1}{\rho }\) has been USED. The higher is the compressibility, the lower the speed of sound.

Correct OPTION is (a) True

To explain: The Prandtl relation is given as a^*2=u1u2 while the characteristic Mach NUMBER is given as M*=\(\frac {u}{a^{*}}\). So, PUTTING this into the Prandtl relation we GET the equation 1=M\(_1^*\)M\(_2^*\) where M\(_1^*\) and M\(_2^*\) are the characteristic Mach number UPSTREAM and downstream of the normal shock.

Correct choice is (c) \(\frac {T_2}{T_1}\)=1

For explanation: A Mach wave is a NORMAL shock wave of diminishing strength. It occurs for M=1 upstream. Then downstream M=1 ALSO. And all the ratios are equal to 1, i.e \(\frac {P_2}{P_1}\)=1, \(\frac {T_2}{T_1}\)=1, \(\frac {\rho_2}{\rho_1}\)=1. This can be found by CALCULATION when we put m=1. The properties across Mach wave do not change.

Correct answer is (a) SPEED of sound

The explanation is: The speed of sound is the quantity that DOMINATES the physical properties of compressible flow. Speed of light is not important in compressible flow PROBLEMS. Density of MEDIUM and distance of propagation are SECONDARY quantities.

Right answer is (b) Mach number only

The EXPLANATION: The remarkable result for the NORMAL shock wave is that for a GIVEN gas (given gamma), the Mach number ahead of the normal shock wave is a function of the Mach number ahead of the normal shock wave only, IRRESPECTIVE of pressure, density, temperature etc. The values are TABULATED and found in gas tables for reference.

Right option is (B) a^*2=u1u2

To elaborate: The Prandtl relation for the normal shock waves taking into account COMBINED form of MASS and MOMENT equation and alternate forms of the energy equation. The FINAL equation comes out in the form of a^*2=u1u2, where u1, u2 are velocities before and after the normal shock.

The correct OPTION is (a) M > 0.3

The EXPLANATION: For the subsonic flows, it depends on the matter of ACCURACY whether to TREAT a flow as compressible or not. For supersonic flows it is always compressible. In general, M > 0.3 can be REGARDED as compressible flow.

The correct ANSWER is (c) a* and a0 are constant ALONG the entire FLOW for an adiabatic, inviscid, steady flow

For explanation: a* and a0 are the defined properties of the flow, constant at a point. They are related to each other but not same. For an adiabatic, inviscid, steady flow they are constant along the streamline. And if all the streamlines are coming from same uniform free-stream, they are constant along the entire flow.