Gas Dynamics, Lecture 3 ( Conservation laws and special flows ) see: astro.ru.nl/~achterb/

1. Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, RadboudUniversiteit Gas Dynamics, Lecture 3(Conservation laws and special flows)see: www.astro.ru.nl/~achterb/

2. Conservative Form of the Equations Aim: To cast all equations in the same generic form: • Reasons: • Allows quick identification of conserved quantities • This form works best in constructing numerical • codes for Computational Fluid Dynamics

3. Generic Form: Transported quantity is a scalar S, so flux F must be a vector! Component form:

4. Generic Form: Transported quantity is a vector M , so the flux must be a tensor T . Component form:

5. Integral properties: Stokes Theorem

6. Examples: mass- and momentum conservation Mass conservation: already in conservation form! Continuity Equation: transport of the scalar Excludes ‘external mass sources’ due to processes like two-photon pair production etc.

7. Example why this is importantfor Numerical Hydrodynamics

8. Fluxes at four cell boundaries! Density inside a cell

9. Examples: mass- and momentum conservation Mass conservation: already in conservation form! Continuity Equation: transport of the scalar Momentum conservation: transport of a vector! Algebraic Manipulation

10. As advertised: Algebraic Manipulation! Starting point: Equation of Motion

11. As advertised: Algebraic Manipulation! • Use: • product rule for differentiation • continuity equation for density

12. As advertised: Algebraic Manipulation!

13. As advertised: Algebraic Manipulation! Use divergence chain rule for dyadic tensors

14. As advertised: Algebraic Manipulation! Rewrite pressure gradient as a divergence

15. As advertised: Algebraic Manipulation!

16. As advertised: Algebraic Manipulation! Stress tensor = momentum flux Momentum source: gravity Momentum density

17. Energy Conservation Energy density is a scalar! Kinetic energy density Internal energy density Gravitational potential energy density Irreversibly lost/gained energy per unit volume

18. Energy Conservation Internal energy per unit mass Specific enthalpy Irreversible gains/losses, e.g. radiation losses “Dynamical Friction”

19. Summary: conservative form of the fluid equations in an ideal fluid: Mass Momentum Energy

20. Entropy conservation (ideal fluid: no heating) ADIABATIC FLUID

21. Special flows Extra mathematical constraints one can put on a flow: Incompressibility: No vorticity (“swirl-free flow”): Steady flow:

22. Example: constant density flow past a sphere

24. Solution:

25. Solution: Far away from sphere: This suggests: m = 1 !

26. Trial Solution:

27. Trial Solution:  A = U

28. Trial Solution:

29. Example: constant density flow past a sphere: solution

30. Flow lines

31. Where has all the non-linearity gone? Constant density flow:

32. It has gone into the pressure! Steady constant-density flow around sphere:

33. For our solution of flow around sphere, on the surface of the sphere:

34. No net pressure forceon the surface of the sphere: NO DRAG PARADOX OF D’ALAMBERT

35. Resolution of the paradox:introduce vorticity!

36. Resolution of the paradox:introduce vorticity! NO fore-aft symmetry, Now there is a drag force!

37. Viscous incompressible flows Viscosity = internal friction due to molecular diffusion, viscosity coefficient : Viscous force density: (incompressible flow!) Equation of motion:

38. Viscosity: diffusive transport ofmomentum

39. Importance of viscosity: the Reynolds Number Re

40. Importance of viscosity: the Reynolds Number Re Very viscous flow:  >> VL, Re << 1 Friction-free flow:  << VL, Re >>1

41. Stokes flow past a sphere (low Reynolds number, Re < 0.2) Because of viscosity: no slip, velocity vanishes on sphere!

42. Step 1: exploit mathematical properties of flow: axisymmetry and incompressibity! Automatically satisfied by writing:

43. Step 2: exploit that the flow is slow due to large viscosity Steady flow equation Slow flow approximation of this equation: From:

44. Step 3: use some vector algebra: Steady slow flow equation Take divergence of slow flow equation:

45. Pressure satisfies harmonic equation:whole books are written on its solution! General solution with constant pressure at infinity:

46. Pressure satisfies harmonic equation:whole books are written on its solution! For this particular case: Components of pressure gradient:

47. Step 2: exploit that the flow is slow due to large viscosity Steady slow flow equation Vorticity:

48. Step 3: use simplified equation of motion: Steady slow flow equation

49. Step 3: use simplified equation of motion: Steady slow flow equation

50. Step 4: solve for ψ(r,θ): Trial solution: