1. Simulation and Animation
Fluids

2. Fluid simulation Content Fluid simulation basics
Terminology
Navier-Stokes equations
Derivation and physical interpretation
Computational fluid dynamics
Discretization
Solution methods

3. Fluid simulation Simulation of the behavior of fluid flow
Interaction and forces between fluid particles and solid bodies
Result of physical properties of fluids
Viscosity generates frictional forces
External forces
Gravitation and other forces
Flow models
Laminar flow
Fluid consists of individual layers sliding over each other
Turbulent flow
Particles in different layers become mixed due to low friction

4. Fluid simulation Flow models
Laminar flow: no fluid exchange between layers
Turbulent flow: no distinct layers, fluid exchange free

5. Fluid simulation Flow movement depends on
Viscous force
Inertial force (Trägheitskraft)
Described by Reynolds number, which depends on
Velocity of the fluid, viscosity  and density , characteristic length D of the flow region
Reynolds number (Re)
The tendency of flow to be laminar (Re is very small) or turbulent (Re is very large)
laminar if Re < 2300
transient for 2300 < Re < 4000
turbulent if Re > 4000
Re =  D V / 

6. Fluid simulation Approaches to describe flow fields Eulerian
Focus is on particular points in the flow occupied by the fluid
Record state of a finite control volume around that point
Dye injection for visualizing flows
Lagrangian
Consider particles and follow their path through the flow
Record state of the particle along the path
Particle tracing for visualizing flows

7. Fluid simulation Basic equations of fluid dynamics Rely on
Physical principles
Conservation of mass
F=ma
Conservation of energy
Applied to a model of the flow
Finite control volume approach
Infinitesimal particle approach
Derivation of mathematical equations
Continuity equation
Navier-Stokes equations

8. Fluid simulation Models of the flow Finite control volume
Control surface S
Control volume V V
Fixed volume, fluid moves through it

9. Fluid simulation Models of the flow Infinitesimal fluid element
Volume dV dV
Fixed fluid element, fluid moves through it
Element moving along the streamlines

10. Flow simulation Governing equations of fluid flow
Finite control volume approach
Apply physical principals to fluid in control volume and passing through control surface
Yields equations in integral form
Distinguish between conservation (fixed volume) and nonconservation (moving volume) form
Infinitesimal fluid element
Apply physical principals to infinitesimal fluid particle
Yiels equations in partial differential form
Distinguish between conservation (fixed particle) and nonconservation (moving particle) form

11. Navier-Stokes equations
A moving fluid element y
V=ui+vj+wk V1
t = 1 x V2
t = 2 z

12. Fluid simulation The time rate of change of density
Particle moves from 1 to 2
Assume density (x,y,z,t) to be continuous and ...
Thus, Taylor series expansion can be performed
Dividing by (t2-t1) ignoring higher order terms and taking the limit:

13. Fluid simulation The substantial derivative
Time rate of change of density when moving from 1 to 2
Physically and numerically different to the time rate of change of density at fixed point (/t) (local derivative)
With

14. Fluid simulation The substantial derivative V: convective derivative
Time rate of change due to movement to position with different properties
D/Dt applied to any variable yields change due to local fluctuations and time and spatial fluctuations
Can be applied to any flow field variable
Pressure (p), temperatur (T), velocity (V) etc.

15. Navier-Stokes equations
The continuity equation
Physical principal: conservation of mass
Finite fixed control volume:
Infinitesimal fluid particle
Net mass flow out of control volume through surface
Time rate of decrease of mass inside control volume =
Net mass flow out of element
Time rate of mass decrease inside element =

16. Navier-Stokes equations
The continuity equation
The model: infinitesimal element fixed in space
Consider mass flux through element dy dz dx

17. Navier-Stokes equations
The continuity equation
Infinitesimal element fixed in space
Net outflow in x-direction (equal for y/z-direction):
Net mass flow:
Time rate of mass increase:

18. Navier-Stokes equations
The continuity equation
Time rate of mass increase inside element
Net mass flow out of element +
The partial differential form of the continuity equation
Other models yield other forms of the continuity equation, which can be obtained from each other

19. Navier-Stokes equations
The momentum equation (Impulsgleichung)
Physical principal: Newton´s second law F=ma
Consider an infinitesimal moving element
Sketch sources of the forces acting on it
Consider x/y/z components separately
Fx = max
First consider left side of F=ma
F = FB + FS
Sum of body forces and surface forces acting on element

20. Navier-Stokes equations
The momentum equation
F = FB + FS (body forces and surface forces)
Body forces
Act at a distance (Gravitational, electric, magnetic forces)
FB = fx (dxdydz)
Surface forces act
Act on surface of element
Can be split into pressure and viscous forces FS = FPress + FVis
Pressure force: imposed by outside fluid, acting inward and normal to surface
Viscous force: imposed by friction due to viscosity, result in shear and normal stress imposed by outside fluid

21. Navier-Stokes equations
Sketch of forces (x-direction only)
Convention: positive increases of V along positive x/y/z Y 6 2
Sy (shear stress)
sz‘ 7 3
P (pressure force) p‘
nx (normal stress)
nx‘ sz 4 X
sy‘ 1 5 Z

22. Navier-Stokes equations
Surface forces
Shear stress
Normal stress Y Y
yx
xx X X
Time rate of change of shear deformation
Time rate of change of shear volume

23. Navier-Stokes equations
Surface forces
On face 0145:
On face 2367:
Equivalent on faces 0246 and 1357 for zx
On face 0123:
On face 4567: X 1 2 3 4 5 6 7
P (pressure force) p‘
Sy (shear stress)
sy‘
nx (normal stress)
nx‘ sz
sz‘

24. Navier-Stokes equations
Total force on fluid element =

25. Navier-Stokes equations
Consider right side of F=ma
Mass of fluid element
Acceleration is time rate of change of velocity (Du/Dt)
Thus (equivalent for v/w):
The Navier-Stokes equations

26. Navier-Stokes equations
What you typically see in the literature is
is the only „strange“ term here
: molecular viscosity
In Newtonian fluids, shear stress is proportional to velocity gradient
[shear stress] =  [strain rate]
Described by Navier-Stokes equations
Non-Newtonian fluids obey different property, e.g. blood, motor oil
Viscosity is not a constant
Depends on temperature and pressure

27. Navier-Stokes equations
From Stokes we know (let‘s just believe it here)
 : molecular viscosity  : second or bulk viscosity

28. Navier-Stokes equations
Incompressible fluids
 = constant
 = constant
 can be taken outside of partial derivatives in NSE
Divergence free (all -terms on previous page vanish)

29. Navier-Stokes equations
With
We obtain
Sketch of derivation:
Write div(V)=0 and resolve for u/x
Partially differentiate both sides with respect to x
Add 2u/x2 on both sides

30. Navier-Stokes equations
Euler equations
Inviscid flow – no viscosity
Only continuity and momentum equation

31. CFD – Computational Fluid Dynamics
Solution methods for governing equations
Governing equations have been derived independent of flow situation, e.g. flow around a car or inside a tube
Boundary (and initial) conditions determine specific flow case
Determine geometry of boundaries and behavior of flow at boundaries
Different kinds of boundary conditions exist
Hold at any time during simulation
Lead to different solutions of the governing equations
Exact solution exists for specific conditions
Initial conditions specify state to start with

32. CFD – Computational Fluid Dynamics
Boundary and initial conditions
Initial conditions should satisfy properties of underlying flow model
E.g. if div(V) = 0 then u=u0(x,y,z), v=v0(x,y,z), w=w0(x,y,z) should satisfy
Boundary conditions specify flow properties at points along the fixed boundary
See below

33. CFD – Computational Fluid Dynamics
Solution methods of partial differential equations
Analytical solutions
Lead to closed-form epressions of dependent variables
Continuously describe their variation
Numerical solutions
Based on discretization of the domain
Replace PDEs and closed form expression by approximate algebraic expressions
Partial derivatives become difference quotients
Involves only values at finite number of discrete points in the domain
Solve for values at given grid points

34. CFD – Computational Fluid Dynamics
Discretization
Layout of grid points on a grid
Location of discrete points across the domain
Arbitrary grids can be employed
Structured or unstructured grids
Implicit or explicit representation of topology (adjacency information)
Uniform grids: uniform spacing of grid points in x and y y
Pi-1j+1
Pij+1
Pi+1j+1 y
Pi-1j
Pij
Pi+1j
y = x
Pi-1j-1
Pij-1
Pi+1j-1 x x

35. CFD – Computational Fluid Dynamics
Finite differences
Approximate partial derivatives by finite differences between points
Derived by considering the Taylor expansion

36. CFD – Computational Fluid Dynamics
Derived by considering the Taylor expansion

37. CFD – Computational Fluid Dynamics
Finite differences for higher order and mixed partial derivatives

38. CFD – Computational Fluid Dynamics
Difference equations - example
The 2D wave equation
Partial Differential Equation
Discretization on a 2D Cartesian grids yields Difference Equation

39. CFD – Computational Fluid Dynamics
Explicit approach
Marching solution with marching variable t
Values at time t+1 are computed from known values at time t and t-1
Step through all interior points of domain and update ut+1

40. CFD – Computational Fluid Dynamics
Explicit vs. Implicit approach
Explicit approach
Difference equation contains one unknown
Can be solved explicitely for it
Implicit approach
Difference equation contains more than one unknown
Solution by simultaneously solving for all unknown
System of algebraic equations to be solved
e.g. Crank-Nicolson

41. CFD – Computational Fluid Dynamics
Implicit Crank-Nicholson scheme

42. Computational Fluid Dynamics
Implicit approach – example
Poisson equation:
Discretization:

43. The Equations Solution of the Navier-Stokes equations Advection
Describes in what
direction a “neighboring”
region of fluid pushes
fluid at u
Diffusion
Describes how quickly variations in velocity are damped-out;
depends on fluid viscosity
Pressure
Describes in which
direction fluid at u is
pushed to reach a
lower pressure area
External Forces
Zero Divergence

44. NSE Discretization

45. Navier-Stokes Equations (cont.)
Rewrite the Navier-Stokes Equations where now F and G can be computed

46. Navier-Stokes Equations (cont.)
Problem: Pressure is still unknown!
From derive:
and end up with this Poisson Equation after discretization:

47. Computational Fluid Dynamics
The algorithm
Step 1: compute Ft and Gt
Use veocities ut and vt and difference equations for partial derivatives
Step 2: solve equations for pressure pt+1
Discretize second order partial derivatives
Use Jacobi, Gauss-Seidel, or Conjugate Gradient method
Step 3: compute new velocities ut+1, vt+1

48. CFD – Computational Fluid Dynamics
Boundary conditions (2D)
No-slip condition
Fluid is fixed to boundary; velocities should vanish at boundaries or have velocities of moving boundaries
Free-slip condition
Fluid is free to move parallel to the boundary; velocity component normal to boundary vanishes
Outflow conditions
Velocity into direction of boundary normal does not change
Inflow conditions
Velocities are given explicitely

49. Demo