Prof dr A Achterberg Astronomical Dept IMAPP Radboud
Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit Gas Dynamics, Lecture 3 (Conservation laws and special flows) see: www. astro. ru. nl/~achterb/
Conservative Form of the Equations Aim: To cast all equations in the same generic form: Reasons: 1. Allows quick identification of conserved quantities 2. This form works best in constructing numerical codes for Computational Fluid Dynamics
Generic Form: Transported quantity is a scalar S, so flux F must be a vector! Component form:
Generic Form: Transported quantity is a vector M , so the flux must be a tensor T. Component form:
Integral properties: Stokes Theorem
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.
Example why this is important for Numerical Hydrodynamics
Fluxes at four cell boundaries! Density inside a cell
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
As advertised: Algebraic Manipulation! Starting point: Equation of Motion
As advertised: Algebraic Manipulation! Use: 1. product rule for differentiation 2. continuity equation for density
As advertised: Algebraic Manipulation!
As advertised: Algebraic Manipulation! Use divergence chain rule for dyadic tensors
As advertised: Algebraic Manipulation! Rewrite pressure gradient as a divergence
As advertised: Algebraic Manipulation!
As advertised: Algebraic Manipulation! Momentum density Stress tensor = momentum flux Momentum source: gravity
Energy Conservation Energy density is a scalar! Kinetic energy density Internal energy density Gravitational potential energy density Irreversibly lost/gained energy per unit volume
Energy Conservation Internal energy per unit mass Specific enthalpy Irreversible gains/losses, e. g. radiation losses “Dynamical Friction”
Summary: conservative form of the fluid equations in an ideal fluid: Mass Momentum Energy
Entropy conservation (ideal fluid: no heating) ADIABATIC FLUID
Special flows Extra mathematical constraints one can put on a flow: 1. Incompressibility: 2. No vorticity (“swirl-free flow”): 3. Steady flow:
Example: constant density flow past a sphere
Solution:
Solution: Far away from sphere: This suggests: m = 1 !
Trial Solution:
Trial Solution: A=U
Trial Solution:
Example: constant density flow past a sphere: solution
Flow lines
Where has all the non-linearity gone? Constant density flow:
It has gone into the pressure! Steady constant-density flow around sphere:
For our solution of flow around sphere, on the surface of the sphere:
No net pressure forceon the surface of the sphere: NO DRAG PARADOX OF D’ALAMBERT
Resolution of the paradox: introduce vorticity!
Resolution of the paradox: introduce vorticity! NO fore-aft symmetry, Now there is a drag force!
Viscous incompressible flows Viscosity = internal friction due to molecular diffusion, viscosity coefficient � : Viscous force density: (incompressible flow!) Equation of motion:
Viscosity: diffusive transport of momentum
Importance of viscosity: the Reynolds Number Re
Importance of viscosity: the Reynolds Number Re Very viscous flow: � >> VL, Re << 1 Friction-free flow: � << VL, Re >> 1
Stokes flow past a sphere (low Reynolds number, Re < 0. 2) Because of viscosity: no slip, velocity vanishes on sphere!
Step 1: exploit mathematical properties of flow: axisymmetry and incompressibity! Automatically satisfied by writing:
Step 2: exploit that the flow is slow due to large viscosity Steady flow equation Slow flow approximation of this equation: From:
Step 3: use some vector algebra: Steady slow flow equation Take divergence of slow flow equation:
Pressure satisfies harmonic equation: whole books are written on its solution! General solution with constant pressure at infinity:
Pressure satisfies harmonic equation: whole books are written on its solution! For this particular case: Components of pressure gradient:
Step 2: exploit that the flow is slow due to large viscosity Steady slow flow equation Vorticity:
Step 3: use simplified equation of motion: Steady slow flow equation
Step 3: use simplified equation of motion: Steady slow flow equation
Step 4: solve for ψ(r, θ): Trial solution:
Step 4, continued
Step 5: apply boundary conditions at sphere/infinity Conditions at infinity:
Step 5: apply boundary conditions at sphere/infinity (continued) Conditions at surface sphere:
Stokes’ solution for viscous flow around sphere: All flow quantities can now be determined:
Surface forces with viscosity
Drag force on sphere
Drag force on sphere For this particular flow at r=a:
- Slides: 57