CP 502 Advanced Fluid Mechanics Compressible Flow Part
CP 502 Advanced Fluid Mechanics Compressible Flow Part 03_Set 01: Stationary normal shock in variable area ducts
Flow through a stationary normal shock wave: Thin normal shock Isentropic upstream; s = sx Mx > 1 My < 1 Isentropic downstream; s = sy > sx Mass, momentum and energy are conserved through a normal shock. Entropy increases across a normal shock. R. Shanthini 21 July 2019
R. Shanthini 21 July 2019
Flow through a stationary normal shock wave: Control volume Isentropic upstream; s = sx Mx > 1 My < 1 Isentropic downstream; s = sy > sx Let us write the balances over the control volume shown, assuming the cross-sectional areas are nearly the same. R. Shanthini 21 July 2019
Balances across a stationary normal shock wave (Rankine-Hugoniot relations): Assuming steady flow across a normal shock assumed to be adiabatic, we get ρx u x = ρy u y = constant (3. 1) = constant (3. 2) = constant (3. 3) (3. 4) R. Shanthini 09 July 21 March 2019 2015
Balances across a stationary normal shock wave (Rankine-Hugoniot relations): Unknowns are: We require one more equation: R. Shanthini 21 July 2019 (3. 5)
Pressure change across stationary normal shock wave: Combining (3. 3) and (3. 5): Using Rearranging: R. Shanthini 21 July 2019 in the above: (3. 6)
Temperature change across stationary normal shock wave: Substituting for in (3. 4): Rearranging: Substituting and rearranging: (3. 7) R. Shanthini 21 July 2019
Density change across stationary normal shock wave: From (3. 1): Using in the above: Using (3. 7) in the above: (3. 8) R. Shanthini 21 July 2019
Mach number change across stationary normal shock wave: Rearranging (3. 5): Substituting from (3. 6), (3. 7) and (3. 8) in the above: R. Shanthini 21 July 2019
Mach number change across stationary normal shock wave: R. Shanthini 21 July 2019
Mach number change across stationary normal shock wave: R. Shanthini 21 July 2019
Mach number change across stationary normal shock wave: Since is a trivial solution, the shock solution is (3. 9) R. Shanthini 21 July 2019
Summary of relationships across stationary normal shock: (3. 6) (3. 7) (3. 8) (3. 9) R. Shanthini 21 July 2019
Pressure ratio in terms of Mx: Combining (3. 6) and (3. 9): (3. 10) R. Shanthini 21 July 2019
Temperature ratio in terms of Mx: Combining (3. 7) and (3. 9): (3. 11) R. Shanthini 21 July 2019
Density ratio in terms of Mx: Combining (3. 5), (3. 10) and (3. 11): (3. 12) R. Shanthini 09 July 21 March 2019 2015
Summary of relationships across stationary normal shock: (3. 6 & 3. 10) (3. 7 & 3. 11) (3. 8 & 3. 12) (3. 9) R. Shanthini 21 July 2019
Stagnation temperature change across stationary normal shock wave: Equation (3. 7) gives the relationship of the temperature change across the shock: (3. 7) Equation (2. 6) relates the temperature to stagnation temperature: (2. 6) Combining the two: R. Shanthini 21 July 2019 (3. 13)
Stagnation pressure change across stationary normal shock wave: Equation (3. 6) gives the relationship of the pressure change across the shock: (3. 6) Equation (2. 7) relates the temperature to stagnation temperature: (2. 7) Combining the two: R. Shanthini 21 July 2019
Stagnation pressure change across stationary normal shock wave: (3. 14) R. Shanthini 21 July 2019
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