Ch6 Differential Conservation Equation for Inviscid Flow

1. Ch6 Differential Conservation Equation for Inviscid Flow 6.1 Introduction Integral Conservation equations → Differential conservation equation • Unsteady flow • Multi-dimensional flow Vector identities :

2. 6.2 Differential Conservation Equation 6.2.1 Continuity Equation ( in Integral Form ) Continuity equation in the differential form ( …diuvergence form) ( …conservatione form) 6.2.2 Momentum Equation Integral form : Momentum equation in the differential form (Euler’s eq)

3. 6.2.3 Energy Equation Integral form : Energy equation in the differential form. Note : the above equation are nonlinear partial differential equations

4. 6.3 The Substantial Derivative (material , Total Derivative) Consider a small fluid element moving through cartesian space, t=t1 t=t2 y y x x z z Lagrangian coordinate system Eulerian coordinate system (Observer is following the fluid particle) (field description) (observer fixed in space) Instaneoue time rater rate of change of density

5. The substantial derivative (material derivative total derivative) Local time derivative Convective derivative

6. 6.4 Differential Conservation Equation in Different Forms 6.4.1 Continuity Equation 6.4.2 Momentum Equation X component - similarly Euler’s equ

7. 6.4.3 Energy equation = Total energy

8. Energy equation in terms of enthelpy h

9. Energy equation in teoms of total enthalpy • The total enthalpy of a moving fluid element in an inviscid flow can be changed due to • Unsteady flow • Heat transfer • Body forces

10. If adiabatic and no body forces If steady furthermore • For an inviscid , adiabatic steady flow with • no body force, ho is const along a given • streamline If ho = const for the flow field ho = const for all the streamline 1st law of thermodynamics The internal and kinetic energy of a moving fluid can be separated such that the 1st law written strictly in terms of internal energy only dose indeed apply to a moving fluid

11. 6.5 The Entropy Equation – the 2nd law of Thermodynamics For a moving fluid particle The entropy equation General for a nonadiabatic viscous flow For an inviscid adiabatic flow The entropy is constant along a streamline in an adiabatic , invisvid flow (for both steady and unsteady flows) Isentropic → (along a streamline) Homentropic → • For compressible flow , the continuity , momentum , and energy equation are sufficients • The entropy equation is used to determine the process direction • However , for isentropic flow s= const may be used to substitute for either the energy or momentum equation

12. 6.6 Crocco’s Theorem – A relation between the Thermodynamics and fluid kinematics of a compressible flow The movement of a fluid element can be both translational and rotational Consider Euler’s eq without body force The 2nd law of thermodnamics Vector identity : Crocco’s theorem (for inviscid flow with no body force)

13. For steady flow vorticity Total enthalpy gradient Gradient of entropy A steady flowfield is rotational if it has gradients of total enthalpy or entropy Consider the flow behind a curved shock wave h01=h02=h0 for all the streamline 3 2 Across the bow shock S1>S2>S3 1 The flowfoeld behind a curved shock is rotational