Fluid dynamics Math background Physics Simulation Related - - PowerPoint PPT Presentation

Fluid dynamics

• Math background
• Physics
• Simulation
• Related phenomena
• Frontiers in graphics
• Rigid fluids

• Domain
• Scalar field
• Vector field

Fields

Ω ⊆ R2 f : Ω → R f : Ω → R2

Types of derivatives

• Derivatives measure how something changes

due to its parameters

• Temporal derivatives
• Spatial derivatives
• gradient operator
• divergence operator
• Laplacian operator

∂f ∂t

∇f = ∂f ∂x, ∂f ∂y T ∇ · f = ∂f x ∂x + ∂f y ∂y ∇2f = ∇ · (∇f) = ∂2f ∂x + ∂2f ∂y

• Math background
• Physics
• Simulation
• Related phenomena
• Frontiers in graphics
• Rigid fluids

Representation

Domain Ω Density ρ : Ω → [0, 1] Velocity u : Ω → R3

“Coffee cup” equation

∂u ∂t = −(u · ∇)u − 1 ρ∇p + s∇2u + f ∇ · u = 0

Navier-Stokes

u: velocity p: pressure s: kinematic viscosity f: body force ρ: fluid density

Momentum equation Incompressibility condition

Momentum equation

• Each particle represents a little blob of fluid

with mass m, a volume V, and a velocity u

• The acceleration of the particle
• The Newton’s law

a ≡ Du Dt mDu Dt = F

Forces acting on fluids

• Gravity: mg
• Pressure: -∇p
• Imbalance of higher pressure
• Viscosity: µ∇·∇u
• Force that makes particle moves at average

speed

• dynamic viscosity coefficient: µ

Momentum equation

ρDu Dt = ρg − ∇p + µ∇ · ∇u mDu Dt = mg − V ∇p + V µ∇ · ∇u Du Dt + 1 ρ∇p = g + s∇ · ∇u

The movement of a blob of fluid Divide by volume Rearrange equation giving Navier-Stoke

Lagrangian v.s. Eulerian

• Lagrangian point of view describes motion as

points travel around space over time

• Eulerian point of view describes motion as the

change of velocity field in a stationary domain

• Lagrangian approach is conceptually easier, but

Eulerian approach makes the spatial gradient easier to compute/approximate

Material derivative

Dq Dt = ∂q ∂t + u ∂q ∂x + v ∂q ∂y + w ∂q ∂z = ∂q ∂t + ∇q · u = Dq Dt d dtq(t, x) = ∂q ∂t + ∇q · dx dt

Advection

• Advection describe how quantity, q, moves

with the velocity field u

• Density
• Velocity

Du Dt = ∂u ∂t + u · ∇u Dρ Dt = ∂ρ ∂t + u · ∇ρ

Advection

∂u ∂t = −(u · ∇)u − 1 ρ∇p + s∇2u + f

Advection

∇ · u = 0

Projection Diffusion Body force

Du Dt + 1 ρ∇p = g + s∇ · ∇u

momentum equation

Density advection

∂ρ ∂t = −(u · ∇)ρ

∂u ∂t = −(u · ∇)u − 1 ρ∇p + s∇2u + f

Velocity advection

s.t. ∇ · u = 0

Projection

∂u ∂t = −(u · ∇)u − 1 ρ∇p + s∇2u + f

Advection

∇ · u = 0

Projection Diffusion Body force

∇ · u > 0? ∇ · u < 0?

Divergence free

∇ ∙ u = 0

Divergence free

∇ ∙ u ≠ 0

Projection

s.t. ∇ · u = 0

∂u ∂t = −(u · ∇)u − 1 ρ∇p + s∇2u + f

Diffusion

∂u ∂t = −(u · ∇)u − 1 ρ∇p + s∇2u + f

Advection

∇ · u = 0

Projection Diffusion Body force

High viscosity fluids

Dropping viscosity

• Viscosity plays a minor role in most fluids
• Numeric methods introduce errors which can

be physically reinterpreted as viscosity

• Fluid with no viscosity is called “inviscid”

Body force

∂u ∂t = −(u · ∇)u − 1 ρ∇p + s∇2u + f

Advection

∇ · u = 0

Projection Diffusion Body force

External forces

• Gravity
• Heat
• Surface tension
• User-specified forces (stirring coffee)

Boundary conditions

• Solid walls
• The fluid cannot go in and out of it
• Control the normal velocity
• Free surfaces
• Air can be represented as a region where

the pressure is zero

• Do not control the velocity at the surface

Solid boundary

• Velocity at the boundary
• No-slip condition:
• Pressure at the boundary
• Make the fluid incompressible AND enforce the

solid boundary condition

u · ˆ n = usolid · ˆ n u = usolid

Physics recap

• Physical quantity represented as fields
• Navier-Stokes PDE describes the dynamics

• Math background
• Physics
• Simulation
• Related phenomena
• Frontiers in graphics
• Rigid fluids

• What is the representation for fluids?
• represent velocity and pressure
• What is the equation of motion for fluids?
• approximate Navier-Stokes in a discrete

domain

• compute Navier-Stokes efficiently

Challenges

Grid structure

pi,j

pi−1,j

pi+1,j pi,j+1 pi,j−1 vi,j+1/2 vi,j−1/2 ui−1/2,j ui+1/2,j

Explicit integration

• Solving for the differential equation explicitly,

namely

• You get this...

∂u ∂t = −(u · ∇)u − 1 ρ∇p + s∇2u + f ut+1 = ut + h ˙ u(t)

Explicit integration

Stable fluids

Invented by Jos Stam Simple, fast, and unconditionally stable

Splitting methods

• Suppose we had a system
• We define simulation function Sf
• Then we could define

∂x ∂t = f(x) = g(x) + h(x) Sf(x, ∆t) : x(t) → x(t) + ∆tf(x) Sf(x, ∆t) : x(t + ∆t) = Sg(x, ∆t) ◦ Sh(x, ∆t)

∂u ∂t = −(u · ∇)u − 1 ρ∇p + s∇2u + f

Splitting methods

w0(x) w1(x) w2(x) w3(x) w4(x) add force Advect Diffuse Project w0 = u(x, t) u(x, t+Δt) = w4

∂u ∂t = −(u · ∇)u − 1 ρ∇p + s∇2u + f

Splitting methods

w0(x) w1(x) w2(x) w3(x) w4(x) add force Advect Diffuse Project w0 = u(x, t) u(x, t+Δt) = w4

Body forces

∂u ∂t = −(u · ∇)u − 1 ρ∇p + s∇2u + f

Splitting methods

w0(x) w1(x) w2(x) w3(x) w4(x) add force Advect Diffuse Project w0 = u(x, t) u(x, t+Δt) = w4

Advection

Numerical dissipation

• Semi-Lagrangian advection tend to smooth out

sharp features by averaging the velocity field

• The numerical errors result in a different

advection equation solved by semi-Largrangian

• Smooth out small vortices in inviscid fluids

∂q ∂t + u ∂q ∂x = u∆x ∂2q ∂x2

∂u ∂t = −(u · ∇)u − 1 ρ∇p + s∇2u + f

Splitting methods

w0(x) w1(x) w2(x) w3(x) w4(x) add force Advect Diffuse Project w0 = u(x, t) u(x, t+Δt) = w4

Diffusion

• Solve for the effect of viscosity
• Use an implicit method for stable result

∂w2 ∂t = ν∇2w2 (I − ν∆t∇2)w3(x) = w2(x)

∂u ∂t = −(u · ∇)u − 1 ρ∇p + s∇2u + f

Splitting methods

w0(x) w1(x) w2(x) w3(x) w4(x) add force Advect Diffuse Project w0 = u(x, t) u(x, t+Δt) = w4

Projection

P( )

Projection

• Projection step subtracts off the pressure from

the intermediate velocity field w3

• project(Δt, u) must satisfies two conditions:
• divergence free:
• boundary velocity:

∇ · ut+1 = 0 ut+1 · ˆ n = usolid · ˆ n w4 = w3 − ∆t1 ρ∇p

Boundary conditions

• Dirichlet boundary condition for free surfaces
• Neumann boundary condition for solid walls

pi+1,j pi,j ui+1/2,j

pi+1,j = pi,j + ρ∆x ∆t (ui+1/2,j − usolid) ut+1

ρ pi+1,j − pi,j ∆x ut+1

since

Divergence-free condition

pi,j

pi−1,j

pi+1,j pi,j+1 pi,j−1

vi,j+1/2 vi,j−1/2 ui−1/2,j ui+1/2,j

ut+1

∆x + vt+1

∆x = 0 ui+1/2,j − ∆t1 ρ pi+1,j − pi,j ∆x

replace ut+1

with result in a discrete Poisson equation

• = −

The pressure equations

• If grid cell (i, j + 1) is in air

and (i+1, j) is a solid

• replace pi,j+1 with zero
• replace pi+1,j with the value

from Neumann condition

pi,j

pi−1,j

pi+1,j pi,j+1 pi,j−1

vi,j+1/2 vi,j−1/2 ui−1/2,j ui+1/2,j

• = −

• pi,j + ρ∆x

Solve a linear system

• Ap = b
• Construct matrix A:
• diagonal entry: # of non-solid neighbors
• off-diagonal: 0 if the neighbor is non-fluid

cell and -1 if the neighbor is fluid cell

• A is symmetric positive definite
• Use preconditioned conjugate gradient